Divergent Series and Differential Equations Michèle Loday-Richaud

Divergent series and differential equations Michèle Loday-Richaud

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Michèle Loday-Richaud. Divergent series and differential equations. 2014. hal-01011050

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mich`ele LODAY-RICHAUD

DIVERGENT SERIES AND DIFFERENTIAL EQUATIONS M. Loday-Richaud LAREMA, Universit´ed’Angers, 2 boulevard Lavoisier 49 045 ANGERS cedex 01 France. E-mail : [email protected] E-mail : [email protected]

2000 Mathematics Subject Classiﬁcation.— M1218X, M12147, M12031. Key words and phrases.— divergent series, summable series, summability, multisummability, linear ordinary diﬀerential equation. DIVERGENT SERIES AND DIFFERENTIAL EQUATIONS

Mich`ele LODAY-RICHAUD

Abstract.— The aim of these notes is to develop the various known approaches to the summability of a class of series that contains all divergent series solutions of ordinary differential equations in the complex field. We split the study into two parts: the first and easiest one deals with the case when the divergence depends only on one parameter, the level k also said critical time, and is called k-summability; the second one provides generalizations to the case when the divergence depends on several (but finitely many) levels and is called multi-summability. We prove the coherence of the definitions and their equivalences and we provide some applications. A key role in most of these theories is played by Gevrey asymptotics. The notes begin with a presentation of these asymptotics and their main properties. To help readers that are not familiar with these concepts we provide a survey of sheaf theory and cohomology of sheaves. We also state the main properties of linear ordinary differential equations connected with the subject we are dealing with, including a sketch algorithm to compute levels and various formal invariants of linear differential equations as well as a chapter on irregularity and index theorems. A chapter is devoted to tangent-to-identity germs of diffeomorphisms in C, 0 as an application of the cohomological point of view of summability.

Prépublications Mathématiques d’Angers Numéro 375 — Janvier 2014

CONTENTS

1. Introduction...... 1

2. Asymptotic expansions in the complex domain...... 5 2.1. Generalities...... 5 2.2. Poincar´easymptotics...... 6 2.3. Gevrey asymptotics...... 17 2.4. The Borel-Ritt Theorem...... 26 2.5. The Cauchy-Heine Theorem...... 31

3. Sheaves and Cechˇ cohomology with an insight into asymptotics 37 3.1. Presheaves and sheaves...... 37 3.2. Cechˇ cohomology...... 54

4. Linear ordinary differential equations: basic facts and infinitesimal neighborhoods of irregular singularities...... 63 4.1. Equation versus system...... 63 4.2. The viewpoint of -modules...... 65 D 4.3. Classifications...... 71 4.4. The Main Asymptotic Existence Theorem...... 87 4.5. Infinitesimal neighborhoods of an irregular singular point...... 90

5. Irregularity and Gevrey index theorems for linear diﬀerential operators...... 99 5.1. Introduction...... 99 5.2. Irregularity after Deligne-Malgrange and Gevrey index theorems . 102 5.3. Wild analytic continuation and index theorems...... 109 2 CONTENTS

6. Four equivalent approaches to k-summability...... 111 6.1. First approach: Ramis k-summability...... 112 6.2. Second approach: Ramis-Sibuya k-summability...... 118 6.3. Third approach: Borel-Laplace summation...... 124 6.4. Fourth approach: wild analytic continuation...... 155

7. Tangent-to-identity diffeomorphisms and Birkhoff Normalisation Theorem...... 159 7.1. Introduction...... 159 7.2. Birkhoff-Kimura Sectorial Normalization...... 162 7.3. Invariance equation of g...... 167 7.4. 1-summability of the conjugacy series h ...... 169

8. Six equivalent approaches to multisummabilitye ...... 171 8.1. Introduction and the Ramis-Sibuya series...... 171 8.2. First approach: asymptotic deﬁnition...... 174 8.3. Second approach: Malgrange-Ramis deﬁnition...... 183 8.4. Third approach: iterated Laplace integrals...... 186 8.5. Fourth approach: Balser’s decomposition into sums...... 195 8.6. Fifth approach: Ecalle’s´ acceleration...... 199 8.7. Sixth approach: wild analytic continuation...... 206

Bibliography...... 211

Index of notations...... 217

Index...... 219 CHAPTER 1

INTRODUCTION

Divergent series may diverge in many various ways. When a divergent series issues from a natural problem it must satisfy specific constraints restricting thus the range of possibilities. What we mean, here, by natural problem is a problem formulated in terms of a particular type of equations such as differential equations, ordinary or partial, linear or non-linear, difference equations, q-difference equations and so on . . . Much has been done in the last decades towards the understanding of the divergence of natural series, their classification and how they can be related to analytic solutions of the natural problem. The question of “summing” divergent series dates back long ago. Famous are the works of Euler and later of Borel, Poincaré, Birkhoff, Hardy and their school until the 1920’s. After a long period of inactivity, the question knew exploding developments in the 1970’s and 1980’s with the introduction by Y. Sibuya and B. Malgrange of the cohomological point of view followed by works of J.-P. Ramis, J. Ecalle´ and many others. In these lecture notes, we focus on the best known class of divergent series, a class motivated by the study of solutions of ordinary linear differential equations with complex meromorphic coefficients at 0 (for short, differential equations) to which they all belong. It is well-known (Cauchy-Lipschitz The- orem) that series solutions of differential equations at an ordinary point are convergent defining so analytic solutions in a neighborhood of 0 in C. At a singular point one must distinguish between regular singular points where all formal solutions are convergent (cf. [Was76, Thm. 5.3] for instance) and irregular singular points where the formal solutions are divergent in general; several examples of divergent series are presented and commented throughout 2 CHAPTER 1. INTRODUCTION the text. The strong point with formal solutions is that they are “easily” computed; at least, there exist algorithms to compute them, whatever the order of the linear differential equation. Nonetheless, one wishes to find actual solutions near such singular points and to understand their behavior. The idea underlying a theory of summation is to build a tool that transforms formal solutions into unique well-defined actual solutions. Roughly speaking, it is natural to ask that the former ones be linked to the latter ones by an asymptotic condition; in other words, that the formal solutions be Taylor series of the actual solutions in a generalized sense. Only convergent series have an asymptotic function on a full neighborhood of 0 in C; otherwise, the asymptotics are required on sectors with vertex 0. Uniqueness is essential to go back and forth and to guaranty good, well-defined properties. The problem is now fully solved for the class under consideration in several equivalent ways providing thus several equivalent theories of summation or theories of summability. Some methods provide necessary and sufficient conditions for a series to be summable, some others provide explicit formulæ. Each method has its own interest; none is the best and their variety must be thought as an enrichment of our means to solve problems. The theories here considered depend on parameters called levels or critical times. The simplest case with only one level k is called k-summability (actually, “simpler than the simplest” is the case when k = 1). The case of several levels k1,k2,...,kν is called multisummability or, to be precise, (k1,k2,...,kν)-summability. At first sight, since the singular points of differential equations are isolated, one could discuss the interest of such a procedure, for, one can approach as close as wished the singular points with the Cauchy-Lipschitz Theorem at the neighboring ordinary points. However, such an approach does not allow a good understanding of the singularities; even numerically, the usual numerical procedures stop being efficient when approaching a singular point, not providing thus even an idea of the behavior at the singular point. On the contrary, a good understanding of the singularity by means of a theory of summation permits a numerical calculation of solutions and of their invariants in most cases.

Chapter 2 deals with asymptotics in the complex domain, ordinary (also called Poincaréasymptotics) and Gevrey asymptotics. The presentation is classical and comes with five examples of divergent series (not all solutions CHAPTER 1. INTRODUCTION 3 of differential equations) that will be commented throughout the text. The chapter contains also a proof of the Borel-Ritt Theorem in Poincaréand in Gevrey asymptotics and a proof of the Cauchy-Heine Theorem in classical form. In chapter 3 we introduce the language of sheaves and rudiments in Cechˇ cohomology. The sheaves , , <0 and ≤−k of germs of various types of A As A A asymptotic functions that are at the core of what follows, are carefully defined. Cohomological versions of the Borel-Ritt Theorem and of the Cauchy-Heine Theorem are made explicit. Chapter 4 contains basic recalls in the theory of ordinary linear differential equations: comparison of equations and systems with Deligne’s Cyclic vector lemma, the viewpoint of -modules, equivalence of equations or systems, for- D mal and meromorphic classifications, Newton polygons and calculation of the formal invariants in the case of equations, Main Asymptotic Existence Theo- rem in sheaf form and in classical form. We end the chapter with the construction of infinitesimal neighborhoods of singularities of differential equations. Chapter 5 is devoted to index theorems for ordinary linear differential operators in various spaces with an application to the irregularity of operators. In chapter 6 we develop four different approaches to k-summability (that is, summability depending on a unique level k) and we prove their equivalence: Ramis k-summability, Ramis-Sibuya k-summability, Borel-Laplace summation with a proof of Nevanlinna’s Theorem and wild-summability, that is, by means of wild analytic continuation in the infinitesimal neighborhood of 0. Follow some applications: Maillet-Ramis Theorem, sufficient conditions for the k- summability of solutions of differential equations, their resurgence in the sense of J. Ecalle,´ and Martinet-Ramis Tauberian Theorems. In each case, we chose the approach that seemed to us to be the most convenient. Chapter 7 deals with tangent-to-identity germs of diffeomorphisms that are formally conjugated to the translation (by 1). It is meant as an application of Ramis-Sibuya Theorem to prove the 1-summability of the conjugacy map. A proof of the Birkhoff-Kimura sectorial normalization Theorem is provided. A careful study by means of Borel and Laplace transforms will be find in [Sau]. In chapter 8 we develop six different approaches to multisummability and we prove their equivalence: an asymptotic definition generalizing Ramis k-summability, Malgrange-Ramis summability generalizing Ramis-Sibuya k- summability, summation by iterated Laplace integrals and accelero-summation 4 CHAPTER 1. INTRODUCTION generalizing the Borel-Laplace summation, Balser’s decomposition into sums and the wild-multisummability in the infinitesimal neighborhood of 0. Some applications to differential equations and Tauberian Theorems are given.

Acknowledgements. I am very indebted to Jean-Pierre Ramis who initiated me to this subject and was always open to my questioning. I also thanks all those that read all or part of the manuscript and especially Anne Duval, Sergio Carillo, Michael Singer, Duncan Sands and Pascal Remy as well as Raymond S´eroul for his “technical” support. CHAPTER 2

ASYMPTOTIC EXPANSIONS IN THE COMPLEX DOMAIN

2.1. Generalities We consider functions of a complex variable x and their asymptotic expansions at a given point x0 of the Riemann sphere. Without loss of generality we assume that x0 = 0 although for some examples classically studied at inﬁnity we keep x = . Indeed, asymptotic expansions at inﬁnity reduce to asymp- 0 ∞ totic expansions at 0 after the change of variable x z = 1/x and asymptotic 7→ expansions at x C after the change of variable x t = x x . The point 0 0 ∈ 7→ − 0 must belong to the closure of the domain where the asymptotics are studied. In general, we consider sectors with vertex 0, or germs of such sectors when the radius approaches 0. The sectors are drawn either in the complex plane C, precisely, in C∗ = C 0 (the functions are then univaluate) or on the \{ } Riemann surface of the logarithm (the functions are multivaluate or given in terms of polar coordinates).

Notations 2.1.1.— We denote by ⊲ = (R) the open sector with vertex 0 made of all points x C α,β ∈ satisfying α< arg(x) <β and 0 < x

′ Deﬁnition 2.1.2.— A sector α′,β′ (R ) is said to be a proper sub-sector of (or to be properly included in) the sector α,β(R) and one denotes

′ α′,β′ (R ) ⋐ α,β(R) 6 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Figure 1

′ ∗ if its closure α′,β′ (R ) in C or is included in α,β(R). ′ ′ ′ Thus, the notation α′,β′ (R ) ⋐ α,β(R) means α < α < β < β and R′

2.2. Poincaréasymptotics Poincaréasymptotic expansions, or for short, asymptotic expansions, are expansions in the basic sense of Taylor expansions providing successive approximations of a function. Unless otherwise mentioned we consider functions of a complex variable and asymptotic expansions in the complex domain and this allows us to use the methods of complex analysis. As we will see, the properties of asymptotic expansions in the complex domain may differ quite a little bit of those in the real domain. In what follows denotes an open sector with vertex 0 either in C∗ or in , the Riemann surface of the logarithm.

2.2.1. Definition. — Definition 2.2.1.— A function f ( ) is said to admit a series n ∈ O n≥0 anx as asymptotic expansion (or to be asymptotic to the series) on a sector if for all proper sub-sector ′ ⋐ of and all N N, there P ∈ exists a constant C > 0 such that the following estimate holds for all x ′: ∈ N−1 f(x) a xn C x N . − n ≤ | | n=0 X ′ The constant C = CN, ′ depends on N and but no condition is required on the nature of this dependence. The technical condition “for all ′ ⋐ ” plays a fundamental role of which we will take benefit soon (cf. Rem. 2.2.10). Observe that the definition includes infinitely many estimates in each of which N is fixed. The conditions have nothing to do with the convergence or 2.2. POINCARE´ ASYMPTOTICS 7 the divergence of the series as N goes to infinity. For N = 1 the condition says that f can be continuously continued at 0 on . For N = 2 it says that the function f has a derivative at 0 on and more generally for any N, that f has a “Taylor expansion” of order N. As in the case of a real variable, asymptotic expansions of functions of a complex variable, when they exist, are unique and they satisfy the same algebraic rules on sums, products, anti-derivatives and compositions. The proofs are similar and we leave them to the reader. The main difference between the real and the complex case lies in the behavior with respect to derivation (cf. Prop. 2.2.9 and Rem. 2.2.10). Notations 2.2.2.— We denote by ⊲ ( ) the set of functions of ( ) admitting an asymptotic expansion A O at 0 on ; ⊲ <0( ) the sub-set of functions of ( ) asymptotic to zero at 0 on . A A Such functions are called flat functions at 0 on ; ⊲ T = T : ( ) C[[x]] the map assigning to each f ( ) its asymp- A → ∈ A totic expansion at 0 on . Due to the uniqueness of the asymptotic expansion, the map T is well defined and is called the Taylor map on (cf. Exa. 2.2.3 below). Due to the algebraic properties of asymptotic expansions the sets ( ) and <0( ) A A are naturally endowed with a structure of vector spaces and the Taylor map is a linear map with kernel <0( ). Proposition 2.2.9 below will improve A this result. We notice that <0( ) is not 0: exponentials of various powers A of x provide examples of non-zero functions of <0( ) for any . For in- A stance, if = x; arg(x) < π/2 , the function exp( 1/x) belongs to <0( ); { | | } − A if = x ; arg(x) < π , the function exp( 1/√x) where √x stands for the { | | } − principal determination of x1/2 belongs to <0( ). A 2.2.2. Examples. — Example 2.2.3 (A trivial example: convergent series) Let be a punctured disc D∗ around 0 (i.e., a sector of opening > 2π in C). If f is an analytic function on D then f is asymptotic to its Taylor series at 0 on D∗. Reciprocally, if f is an analytic function on D∗ that has an asymptotic expansion at 0 on D∗ then, f is bounded near 0 and according to the removable singularity Theorem, f is analytic on all of D. 8 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Example 2.2.4 (A fundamental example: the Euler function)

Consider the Euler equation dy (1) x2 + y = x. dx Looking for a power series solution one finds the unique series (2) E(x)= ( 1)n n! xn+1 n≥0 − called the Euler series. The Euler seriesX is clearly divergent for all x = 0 and thus, it does e 6 not provide an analytic solution near 0 by Cauchy summation. However, an actual solution can be found by applying the Lagrange method on R+; notice that 0 is a singular point of the equation and the Lagrange method must be applied on a domain (i.e., a connected open set) containing no singular point (R+ is connected, open in R and does not contain 0). Among the infinitely many solutions given by the method we choose the only one which is bounded as x tend to 0+; it reads x 1 1 dt +∞ e−ξ/x E(x)= exp + = dξ 0 − t x t 0 1+ ξ Z Z and is not only a solution on R+ but also a well defined solution on (x) > 0. ℜ Actually, the function E is asymptotic to the Euler series E on x C ; (x) > 0 . { ∈ ℜ } A proof works as follows: writing N−2 e 1 ξN−1 = ( 1)nξn +( 1)N−1 1+ ξ − − 1+ ξ n=0 X +∞ n −u and using 0 u e du = Γ(1+ n), we get the relation N−2 R +∞ ξN−1 e−ξ/x E(x)= ( 1)n Γ(1 + n) xn+1 +( 1)N−1 dξ − − 1+ ξ n=0 0 X Z and we are left to bound the integral remainder term. Choose 0 <δ<π/2 and consider the (unlimited) proper sub-sector

δ = x ; arg(x) < π/2 δ | | − of the half-plane = x ; (x) > 0 . { ℜ }

Figure 2 2.2. POINCARE´ ASYMPTOTICS 9

For all x δ, we can write ∈ N−2 +∞ E(x) ( 1)n n! xn+1 ξN−1 e−ℜ(ξ/x) dξ − − ≤ n=0 0 X Z +∞ ξN−1 e−ξ sin(δ)/|x| dξ ≤ Z0 x N +∞ = | | uN−1 e−u du = C x N (sin δ)N | | Z0 with C = Γ(N)/(sin δ)N . This proves that the function E(x) is asymptotic to the Euler series E(x) at 0 on the half plane . Observe that the constant C does not depend on x but it depends on N and δ and it tends to inﬁnity as δ tends to 0. Thus, the estimate is no longere valid on the whole sector = x ; (x) > 0 . { ℜ } If we slightly turn the line of integration to the line dθ with argument θ then, the same calculation stays valid and provides a function Eθ(x) with the same asymptotic expansion on the half plane bisected by dθ. Due to Cauchy’s Theorem, Eθ(x) is the analytic continuation of E(x). Denote by E(x) the largest analytic continuation of the initial function E(x) by such a method. Its domain of deﬁnition is easily determined: we can rotate the line dθ as long as it does not meet the pole ξ = 1 of the integrand, i.e., we − can rotated it from θ = π to θ =+π. We get so an analytic continuation of the initial − function E on the sector

E = x ; 3π/2 < arg(x) < +3π/2 { ∈ − } of the universal cover of C∗. On such a sector, E(x) is asymptotic to the Euler series E(x).

Figure 3

With this construction we are given on x C∗ ; (x) < 0 two determinations E+(x) { ∈ ℜ } and E−(x) of E(x) when the direction θ approaches +π and π respectively. Let us − observe the following two facts: ⊲ The determinations E+(x) and E−(x) are distinct since, otherwise, the function E(x) would be analytic at 0 and the Euler series E(x) would be convergent. ⊲ Although E(x) admits an analytic continuation as a solution of the Euler equation ∗ e on all of the universal cover of C (Cauchy-Lipschitz Theorem) its stops having an asymptotic expansion on any sector larger than E (i.e., E ( ). Indeed, the two determinations E+(x) and E−(x) satisfy the relation (see [LR90] or the calculation of the variation of E(x) in Remark 2.5.3) (3) E+(x) E−(x)=2πi e1/x. − 10 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Thus, E+(x) can be continued through the negative imaginary axis by setting E+(x)=E(x)+2πie1/x and symmetrically for E−(x) through the positive imaginary axis. Any asymptotic condition fails since e1/x is unbounded at 0 when (x) is positive. ℜ Such a phenomenon of discontinuity of the asymptotics is called Stokes phenomenon (see end of Rem. 2.5.3 and Sect. 4.3). The function E(x) is called the Euler function. Unless otherwise specified we consider it as a function defined on x ; arg(x) < 3π/2 . { ∈ | | } Example 2.2.5 (A classical example: the exponential integral) The exponential integral Ei(x) is the function given by +∞ dt (4) Ei(x)= e−t t · Zx The integral being well defined on horizontal lines avoiding 0 the function Ei(x) is well defined and analytic on the plane C slit along the real non positive axis. Let us first determine its asymptotic behavior at the origin 0 on the right half plane = x ; (x) > 0 . For this, we start with the asymptotic expansion of its deriva- { ℜ } tive Ei′(x)= e−x/x. Taylor expansion with integral remainder for e−x gives − N−1 xn xN 1 e−x = ( 1)n +( 1)N (1 u)N−1 e−ux du − n! − (N 1)! − n=0 0 X − Z and then, since ( ux) < 0, ℜ − N−1 1 xn−1 x N−1 Ei′(x)+ + ( 1)n | | x − n! ≤ N! · n=1 X We see that a negative power of x occurs with a logarithm as anti-derivative. Integrating between ε> 0 and x and making ε tend to 0 we obtain N−1 xn x N Ei(x)+ln(x)+ γ + ( 1)n | | with γ = lim Ei(x)+ln(x) . − n n! ≤ N! − x→0+ n=1 · X To fit our definition of an asymptotic expansion we must consider the function Ei(x)+ln(x). By extension, one says that Ei(x) has the asymptotic expansion ∞ xn ln(x) γ ( 1)n − − − − n n! · n=1 X · We leave as an exercise the fact that γ is indeed the Euler con- n stant limn→+∞ 1/p ln(n). Notice that, this time, we did not need to shrink the p=1 − sector . P Look now what happens at infinity. Instead of calculating the asymptotic expansion of Ei(z) at infinity from its definition we notice that the function y(x)= e1/x Ei(1/x) is the Euler function f(x). Hence, it has on at 0 the same asymptotics as f(x). Turning back to the variable z =1/x we can state that ez Ei(z) has the series ( 1)n n!/zn+1 ≃∞ n≥0 − as asymptotic expansion at infinity on . By extension, one says that Ei(z) is asymptotic P to e−z ( 1)nn!/zn+1 on at infinity. n≥0 − P 2.2. POINCARE´ ASYMPTOTICS 11

Example 2.2.6 (A generalized hypergeometric series 3F0)

We consider a generalized hypergeometric equation with given values of the parameters, say, d d d d (5) D3,1y z z +4 z z +1 z 1 y =0. ≡ dz − dz dz dz − The equation has an irregular singular point at inﬁnity and a unique series solution 1 (n + 2)!(n + 3)!(n + 4)! 1 (6) g(z)= z4 2!3!4!n! zn · nX≥0 Using the standard notatione for the hypergeometric series, the series g reads

−4 1 g(z)= z 3F0 3, 4, 5 . { } z e By abuse of language, we will also call g an hypergeometric series. e One can check that the equation admits, for 3π < arg(z) < +π, the solution e − 1 g(z)= Γ(1 s)Γ( s)Γ( 1 s)Γ(4 + s)eiπszs ds 2πi 2!3!4! − − − − Zγ where γ is a path from 3 i to 3+ i along the line (s)= 3. This follows from − − ∞ − ∞ ℜ − the fact that the integrand G(s, z) satisﬁes the order one diﬀerence equation deduced from

D3,1 by applying a Mellin transform. We leave the proof to the reader. Instead, let us check that the integral is well defined. The integrand G(s, z) being continuous along γ we just have to check the behavior of G(s, z) as s tends to infinity along γ. An asymptotic expansion of Γ(t + iu) for t R fixed and u R large is given by (see [BH86, p. 83]): ∈ ∈ 1 π π 1 (7) Γ(t + iu)= u t− 2 e− 2 |u| √2π ei 2 (t− 2 ) sgn(u)−iu u iu 1+ O 1/u . | | | | It follows that G(t + iu, z) satisfies (8) G(t + iu, z) = (2π)2 u −2t+2 z t e−2π|u|−πu−u arg(z) 1+ O(1/u) . | | | | The exponent of the exponential being negative for 3π < arg( z) < π the integral is − convergent and it defines an analytic function g(z). Let us prove that the function g(z) is asymptotic to g(z) at infinity on the sector = z ; 3π < arg(z) < +π . For this, consider a path { − } ∗e γn,p = γ1 γ2 γ3 γ4 (n,p N ) ∪ ∪ ∪ ∈ as drawn on Fig.4.

The path γn,p encloses the poles sm = 4 m for m =0,...,n + 1 of G(s, z) and the − − −4−m residues are Res G(s, z); s = 4 m = (2+ m)!(3 + m)!(4 + m)! z /m! = 2!3!4! am. − − Indeed, Γ(4 + s) has a simple pole at s = 4 m and reads − − ( 1)m Γ 4+( 4 m + t) = Γ( m + t)= − t−1 + O(1) − − − m! while all other factors of G are non-zero analytic functions. We deduce that n+1 1 1 a G(s, z)ds = m 2πi 2!3!4! z4 zm · γn,p m=0 Z X 12 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Figure 4. Path γn,p

Formula (8) implies the estimate G(t + iεp, z) Cp2n+5 e−(2π+επ+ε arg(z))p, ε = 1, ≤ ± valid for z > 1 all along γ2 γ4 , the constant C depending on n and z but not on p. This | | ∪ shows that the integral along γ2 γ4 tends to zero as p tends to inﬁnity and consequently, ∪ we obtain n+1 1 a g(z)= m + g (z) z4 zm n m=0 1 Xn 3 where gn(z) = G(s, z)ds and γ = s C ; (s) = 4 n oriented 2πi 2!3!4! γn { ∈ ℜ − − − 2 } upwards. R For any (small enough) δ > 0 consider the proper sub-sector δ of deﬁned by

δ = z C ; z > 1 and 3π + δ < arg(z) <π δ . ∈ | | − − 3 n s For z δ and s = 4 n + iu γ , the factor z satisﬁes ∈ − − − 2 ∈ −u(π−δ) s 1 e if u< 0, z 3 u(3π−δ) | | ≤ z 4+n+ 2 · e if u> 0. | | and using again formula (7) we obtain Const G 4 n 3 + iu, z n,δ u 13+n e−|u|δ. − − − 2 ≤ z (4+n)+1 | | | | Hence, there exists a constant C = C(n,δ) depending on n and δ but not on z such that n+1 1 am C g(z) = gn(z) for all z δ. − z4 zm ≤ z (4+n)+1 ∈ m=0 | | X Rewriting this estimate in the form n 1 am an+1 C + an+1 g(z) = gn(z)+ | | for all z δ − z4 zm z(4+n)+1 ≤ z (4+n)+1 ∈ m=0 | | X we satisfy Deﬁnition 2.2.1 for g at the order 4+ n . With this method we do not know how the constant C depends on n but we know 2 2 that an+1 grows like (n!) and then C + an+1 itself grows at least like (n!) . | | | | 2.2. POINCARE´ ASYMPTOTICS 13

Example 2.2.7 (A series solution of a mild diﬀerence equation)

Consider the order one diﬀerence equation 1 (9) h(z + 1) 2h(z)= − z · A diﬀerence equation is said to be mild when its companion system, here

y1(z + 1) 2 1/z y1(z) = " y2(z + 1) # " 0 1 #" y2(z) # 2 0 has an invertible leading term; in our case, 0 1 is invertible. The term “mild” and its contrary “wild” were introduced by M. van der Put and M. Singer [vdPS97]. Let us look at what happens at infinity. By identification, we see that equation (9) n has a unique power series solution in the form h(z) = n≥1 hn/z . The coefficients hn are defined by the recurrence relation e P p (m + p 1)! hn = ( 1) hm − − (m 1)! p! m+p=n − m,pX≥1

starting from the initial value h1 = 1. It follows that the sequence hn is alternate and − satisﬁes

hn n hn−1 . | | ≥ | | Consequently, hn n! and the series h is divergent. Actually the recurrence relation can | | ≥ be solved as follows. Consider the Borel transform e ζn−1 h(ζ)= h n (n 1)! nX≥1 − b of h (cf. Def. 6.3.1). It satisﬁes the Borel transformed equation e−ζ h(ζ) 2h(ζ)=1 and − then h(ζ)=1/(e−ζ 2). Its Taylor series at 0 reads − e n+1 n b b ( 1) p n b T0h(ζ)= − ζ n! 2p+1 nX≥0 Xp≥0 nb n−1 p+1 which implies that hn =( 1) p≥0 p /2 . Again, we see that the series h is diver- n−1 n−+1 gent since hn n /2 . | | ≥ P e We claim that the function +∞ h(z)= h(ζ)e−zζ dζ Z0 is asymptotic to h(z) at inﬁnity on the sector b= z ; (z) > 0 (right half-plane). Indeed, {π ℜ } π choose N N and a proper sub-sector δ = z ; + δ < arg(z) < δ of . From the ∈ { − 2 2 − } Taylor expansione with integral remainder of h(ζ) at 0

N n−1 N 1 ζ ζ N−1 (N) h(ζ)= hn + b (1 t) h (ζt)dt (n 1)! (N 1)! − n=1 0 X − − Z b b we obtain N h +∞ ζN 1 h(z)= n + (1 t)N−1 h(N)(ζt)dt e−zζ dζ. zn (N 1)! − n=1 0 0 X Z − Z b 14 CHAPTER 2. ASYMPTOTIC EXPANSIONS

To bound h(N)(ζt) we use the Cauchy Integral Formula N! h(u) b h(N)(ζt)= du 2πi (u ζt)N+1 Cζt Z −b where Cζt denotes the circleb with center ζt, radius 1/2, oriented counterclockwise. For t [0, 1] and ζ [0, + [ then ζt is non negative and (u) 1/2 when u runs ∈ ∈ ∞ ℜ ≥ − over any Cζt. Hence, we obtain N!2N 1 (N 1)!2N h(N)(ζt) and (1 t)N−1h(N)(ζt)dt − ≤ 2 e1/2 − ≤ 2 e1/2 · Z0 − − Finally, from the identity above we can conclude that, for all z δ, b b ∈ N h 2N +∞ C (10) h(z) n ζN e−ζz dζ = − zn ≤ 2 e1/2 z N+1 n=1 Z0 X − | | N with C = 1 N! 2 . 2− e1/2 (sin δ)N+1 This proves that h(z) is asymptotic to h(z) at inﬁnity on .

Example 2.2.8 (A series solution of a wild diﬀerence equation)

Consider the order one inhomogeneous wild difference equation 1 1 1 (11) ℓ(z +1)+ 1+ ℓ(z)= z z z · An identification of terms of equal power shows that it admits a unique series solution −n ℓ(z)= ℓnz nX≥1 whose coefficients ℓn are given by thee recurrence relation

p (m + p 1)! ℓn+1 = 2ℓn ( 1) ℓm − − − − p!(m 1)! m+p=n − m≥X1,p≥1

from the initial value ℓ1 = 1. It follows that the sequence (ℓn)n≥1 is alternate and satisﬁes

ℓn+1 (n 1) ℓn−1 . | | ≥ − | | n Hence, ℓ2n 2 (n 1)! for all n and consequently, the series is divergent. The Borel ≥ − transform ℓ(ζ) of the series ℓ(z) satisﬁes the equation ζ ζ b e e−ξℓ(ξ)dξ + ℓ(ξ)dξ + ℓ(ξ)=1 Z0 Z0 equivalent to the two conditions ℓb(0) = 1 and ℓb′(ζ)= b e−ζ 1 ℓ(ζ). Hence, − − 1 −ζ ℓ(ζ)= e−ζ+e . b e b b +∞ −zζ We leave as an exercise to prove thatb the Laplace integral 0 ℓ(ζ)e dζ is a solution of (11) asymptotic to ℓ(z) at inﬁnity on the sector (z) > 1 (Follow the same method ℜ R− as in the previous exercise and estimate the constant C). b b 2.2. POINCARE´ ASYMPTOTICS 15

2.2.3. Algebras of asymptotic functions. — Recall that denotes a given open sector with vertex 0 in C 0 or in the Riemann surface of the \{ } logarithm . Unless otherwise mentioned we refer to the usual derivation d/dx and to Notations 2.2.2.

Proposition 2.2.9 (Diﬀerential algebra and Taylor map)

⊲ The set ( ) endowed with the usual algebraic operations and the usual A derivation d/dx is a diﬀerential algebra. ⊲ The Taylor map T = T : ( ) C[[x]] is a morphism of diﬀerential A → algebras with kernel <0( ). A

Proof. — Due to the algebraic rules on asymptotic expansions ( ) is a sub- A algebra of ( ). We are left to prove that ( ) is stable under derivation with O A respect to x and that the Taylor map T commutes with derivation. Let f ( ) have an asymptotic expansion T f(x)= a xn. Since f ∈ A n≥0 n belongs to ( ) it admits a derivative f ′ ( ). Moreover, for all ′ ⋐ and O ∈O P all N 0, there exists C > 0 such that, for all x ′, ≥ ∈ N f(x) a xn C x N+1 − n ≤ | | n=0 X and we want to prove that for all ′′ ⋐ , for all N > 0, there exists C′ > 0 such that, for all x ′′, ∈ N−1 f ′(x) (n + 1)a xn C′ x N . − n+1 ≤ | | n=0 X

Fix N > 0 and consider the function g(x)= f(x) N a xn. − n=0 n We must prove that the condition P for all ′ ⋐ , there exists C > 0 such that g(x) C x N+1 for all x ′ • | | ≤ | | ∈ implies the condition for all ′′ ⋐ , there exists C′ > 0 such that g′(x) C′ x N for all x ′′. • | | ≤ | | ∈ Given ′′ ⋐ , choose a sector ′ such that ′′ ⋐ ′ ⋐ (see Fig 5) and let δ be so small that, for all x ′′, the closed disc B(x, x δ) centered at x with ∈ | | radius x δ be contained in ′. Denote by γ the boundary of B(x, x δ). | | x | | 16 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Figure 5

By assumption, for all t B(x, x δ) and, especially, for all t γ the ∈ | | ∈ x function g satisﬁes g(t) C t N+1. We deduce from Cauchy’s integral for- ′ 1 | g(t)| ≤ | | ′′ ′ mula g (x)= 2 dt that, for all x , the derivative g satisﬁes 2πi γx (t−x) ∈ ′ 1R 2π x δ C N+1 ′ N g (x) max g(t) | | 2 x (1 + δ) = C x ≤ 2π t∈γx ( x δ) ≤ x δ | | | | | | | | with C ′ = C (1 + δ)N+1/δ . Hence, the result.

Remarks 2.2.10. — Let us insist on the role of Cauchy’s integral formula. ⊲ The proof of Proposition 2.2.9 does require that the estimates in Defi- nition 2.2.1 be satisfied for all ′ ⋐ instead of itself. Otherwise, we could not apply Cauchy’s integral formula and we could not assert anymore that the algebra ( ) is differential. In such a case, algebras of asymptotic functions A would not be suitable to handle solutions of differential equations. ⊲ Theorem 2.2.9 is no longer valid in real asymptotics, where Cauchy’s integral formula does not hold, as it is shown by the following counter-example. The function f(x)= e−1/x sin(e1/x) is asymptotic to 0 (the null series) on R+ at 0. Its derivative f ′(x) = 1 e−1/x sin(e1/x) 1 cos(e1/x) has no x2 − x2 limit at 0 on R+ and then no asymptotic expansion. This proves that the set of real analytic functions admitting an asymptotic expansion at 0 on R+ is not a differential algebra.

The following proposition provides, in particular, a proof of the uniqueness of the asymptotic expansion, if any exists.

Proposition 2.2.11.— A function f belongs to ( ) if and only if f belongs A to ( ) and a sequence (a ) exists such that O n n∈N 1 (n) ′ lim f (x)= an for all ⋐ . n! x→0 x∈ ′ 2.3. GEVREY ASYMPTOTICS 17

Proof. — The only if part follows from Proposition 2.2.9. To prove the if part consider ′ ⋐ . For all x and x ′, f admits the Taylor expansion 0 ∈ with integral remainder N−1 1 x 1 f(x) f (n)(x )(x x )n = (x t)N−1f (N)(t)dt. − n! 0 − 0 (N 1)! − n=0 x0 X Z − Notice that we cannot write such a formula for x0 = 0 since 0 does not even belong to the deﬁnition set of f. However, by assumption, the limit of the left ′ hand side as x0 tends to 0 in exists; hence, the limit of the right hand side exists and we can write N−1 x 1 f(x) a xn = (x t)N−1f (N)(t)dt. − n (N 1)! − n=0 0 X Z − Then, N−1 f(x) a xn 1 x(x t)N−1f (N)(t)dt − n ≤ (N−1)! 0 − n=0 X N R |x| (N) N sup ′ f (t) C x , ≤ N! t∈ ≤ | | 1 (N) the constant C = sup ′ f (t) being ﬁnite by assumption. Hence, the N! t∈ | | conclusion

2.3. Gevrey asymptotics When working with differential equations for instance, it appears easily that the conditions required in Poincaréasymptotics are too weak to fit some natural requests, say for instance, to provide asymptotic functions that are solutions of the equation when the asymptotic series themselves are solution or, better, to set a 1-to-1 correspondence between the series solution and their asymptotic expansion. A precise answer to these questions is found in the theories of summation (cf. Chaps. 6 and 8). A first step towards that aim is given by strengthening Poincaréasymptotics into Gevrey asymptotics. From now on, we are given k > 0 and we denote its inverse by

s = 1/k

When k > 1/2 then π/k < 2π and the sectors of the critical opening π/k to be further considered may be seen as sectors of C∗ itself; otherwise, they must be considered as sectors of the universal cover of C∗. In general, 18 CHAPTER 2. ASYMPTOTIC EXPANSIONS depending on the problem, we may assume that k > 1/2 after performing a change of variable (ramiﬁcation) x = tp with a large enough p N. ∈ Recall that, unless otherwise speciﬁed, we denote by , ′,... open sectors in C∗ or and that the notation ′ ⋐ means that the closure of the sector ′ in C∗ or lies in (cf. Def. 2.1.2).

2.3.1. Gevrey series. — Deﬁnition 2.3.1 (Gevrey series of order s or of level k)

n A series n≥0 anx is of Gevrey type of order s (in short, s-Gevrey) if there exist constants C > 0,A> 0 such that the coeﬃcients a satisfy P n a C(n!)sAn for all n. | n| ≤ The constants C and A do not depend on n. n n s Equivalently, a series n≥0 anx is s-Gevrey if the series n≥0 anx /(n!) converges. P P Notation 2.3.2.— We denote by C[[x]]s the set of s-Gevrey series.

Observe that the spaces C[[x]]s are ﬁltered as follows:

C x = C[[x]] C[[x]] C[[x]] ′ C[[x]] = C[[x]] { } 0 ⊂ s ⊂ s ⊂ ∞ for all s,s′ satisfying 0

⊲ A convergent series (cf. Exa. 2.2.3) is a 0-Gevrey series. ⊲ The Euler series E(x)(cf. Exa. 2.2.4) is 1-Gevrey and hence s-Gevrey for any s> 1. It is s-Gevrey for no s< 1. e ⊲ The hypergeometric series 3F0(1/z) (cf. Exa. 2.2.6) is 2-Gevrey and s-Gevrey for no s< 2. ⊲ The series h(z) (cf. Exa. 2.2.7) is 1-Gevrey. Indeed, it is at least 1-Gevrey

since hn n! and it is at most 1-Gevrey since its Borel transform at inﬁnity converges. | | ≥ e n n ⊲ From the fact that ℓ2n+1 2 n! and ℓ2n 2 (n 1)! we know that, if the | | ≥ | | ≥ − series ℓ˜(z) (cf. Exa. 2.2.8) is of Gevrey type then it is at least 1/2-Gevrey. From the fact that its Borel transform is convergent it is of Gevrey type and at most 1-Gevrey. Note however that its Borel transform is an entire function and consequently, ℓ˜(z) could be less than 1-Gevrey.

Proposition 2.3.4.— C[[x]]s is a diﬀerential sub-algebra of C[[x]] stable under composition. 2.3. GEVREY ASYMPTOTICS 19

Proof.— C[[x]]s is clearly a sub-vector space of C[[x]]. We have to prove that it is stable under product, derivation and composition.

⊲ Stability of C[[x]]s under product. — Consider two s-Gevrey series n n n≥0 anx and n≥0 bnx satisfying, for all n and for positive constants A, B, C and K, the estimates P P a C(n!)sAn and b K(n!)sBn. | n| ≤ | n| ≤ n Their product is the series n≥0 cnx where cn = p+q=n apbq. Then, P P c CK (p!)s(q!)sApBq CK(n!)s(A + B)n. | n| ≤ ≤ p+q=n X Hence the result.

⊲ Stability of C[[x]]s under derivation. — Given an s-Gevrey series a xn satisfying a C(n!)sAn for all n, its derivative b xn n≥0 n | n| ≤ n≥0 n satisfies P P b =(n + 1) a (n + 1)C((n + 1)!)sAn+1 C′(n!)sA′n | n| | n+1| ≤ ≤ for convenient constants A′ > A and C′ C. Hence the result. ≥ ⊲ Stability of C[[x]]s under composition [Gev18]. — Let f(x) = p n p≥1 apx and g(y)= n≥0 bny be two s-Gevrey series. The composition g f(x) = c xn provides a well-defined power series in xe. From P ◦ n≥0 Pn the hypothesis, theree exist constants h,k,a,b > 0 such that, for all p and n, P s p the coefficientse e of the series f and g satisfy respectively ap h(p!) a and | | ≤ b k(n!)sbn. | n| ≤ Faàdi Bruno’s formula allowse use to write n mj n! c = N(m) m ! b j! a n | | |m| j mX∈In jY=1 where In stands for the set of non-negative n-tuples m = (m1, m2, . . . , mn) satisfying the condition n jm = n, where m = n m and the coeffi- j=1 j | | j=1 j cient N(m) is a positive integer depending neither on f nor on g. Using the P n P Gevrey hypothesis and the condition j=1 jmj = n, we can then write e e n P 1+s n! c kan N(m) m !1+s (hb)|m| j! mj . | n| ≤ | | mX∈In jY=1 20 CHAPTER 2. ASYMPTOTIC EXPANSIONS

As clearly m n and N(m) N(m)1+s, with B = max(hb, 1), we obtain | | ≤ ≤ n 1+s n! c k (aB)n N(m) m ! j! mj | n| ≤ | | mX∈In jY=1 and then, from the inequality K X1+s K X 1+s for non-negative s i=1 i ≤ i=1 i and X ’s, the estimate i P P n 1+s n! c k (aB)n N(m) m ! j! mj . | n| ≤ | | mX∈In jY=1 Now, applying Fa`adi Bruno’s formula to the case of the series f(x)= x/(1 x) − and g(x) = 1/(1 x), implying thus g f(x)=1+ x/(1 2x), we get the − ◦ − relation e n e2n−1n! when n 1 e N(m) m ! (j!)mj e= ≥ | | 1 when n = 0; mX∈In jY=1 hence, a fortiori, n N(m) m ! (j!)mj 2n n! | | ≤ mX∈In jY=1 and we can conclude that c k (n!)s (21+s aB)n | n| ≤ for all n N, which ends the proof. ∈ One has actually the more general result stated in Proposition 2.3.6 below.

Deﬁnition 2.3.5.— A series g(y ,...,y )= b yn1 ...ynr 1 r n1,··· ,nr≥0 n1,...,nr 1 r is said to be (s ,...,s )-Gevrey if there exist positive constants C,M ,...,M 1 r P 1 r such that, for all n-tuple (n1,...,ne r) of non-negative integers, the series sat- isﬁes an estimate of the form b C(n !)s1 (n !)sr M n1 M nr . n1,...,nr ≤ 1 ··· r 1 ··· r

It is said to be s-Gevrey when s1 = = sr = s. ··· Proposition 2.3.6.— Let f1(x), f2(x),..., fr(x) be s-Gevrey series without constant term and let g(y1,...,yr) be an s-Gevrey series in r variables. Then, the series g(f1(x),...,efr(x))eis an s-Gevreye series. e th Since the expression of the n derivative of g(f1(x),..., fr(x)) has the e e e same form as in the case of g(f(x)) the proof is identical to the one for g(f(x)) and we leave it as an exercise. e e e e e e e 2.3. GEVREY ASYMPTOTICS 21

The result is, a fortiori, true when g or some of the fj’s are analytic. The fact that C[[x]]s be stable by product (and composition of course) can then be seen as a consequence of that proposition.e e

2.3.2. Algebras of Gevrey asymptotic functions. —

Deﬁnition 2.3.7 (Gevrey asymptotics of order s) A function f ( ) is said to be Gevrey asymptotic of order s (for short, ∈O n s-Gevrey asymptotic) to a series n≥0 anx on if for any proper sub-sector ′ ⋐ there exist constants C ′ > 0 and A ′ > 0 such that, the following P estimate holds for all N N∗ and x ′: ∈ ∈ N−1 n s N N (12) f(x) a x C ′ (N!) A ′ x − n ≤ | | n=0 X A series which is the s-Gevrey asymptotic expansion of a function is said to be an s-Gevrey asymptotic series.

Notation 2.3.8.— We denote by ( ) the set of functions admitting an As s-Gevrey asymptotic expansion on .

1 Given an open arc I of S , let I (R) denote the sector based on I with radius R. Since there is no possible confusion, we also denote the set of germs of functions admitting an s-Gevrey asymptotic expansion on a sector based on I by (I)= lim (R) . s −→ s I A R→0 A ′ ∗ The constants C ′ and A ′ may depend on ; they do not depend on N N ∈ and x ′. Gevrey asymptotics diﬀers from Poincar´easymptotics by the fact ∈ that the dependence on N of the constant CN, ′ (cf. Def. 2.2.1) has to be of Gevrey type.

Comments 2.3.9 (On the examples of chapter 1)

The calculations in Section 2.2.2 show the following Gevrey asymptotic properties: ⊲ The Euler function E(x) is 1-Gevrey asymptotic to the Euler series E(x) on any

(germ at 0 of) half-plane bisected by a line dθ with argument θ such that π<θ< +π. − It is then 1-Gevrey asymptotic to E(x) at 0 on the full sector 3π/2 < arg(xe) < +3π/2. − ⊲ Up to an exponential factor the exponential integral has the same properties on germs of half-planes at inﬁnity. e 22 CHAPTER 2. ASYMPTOTIC EXPANSIONS

⊲ The generalized hypergeometric series g(z) of Example 2.2.6 is 2-Gevrey and we stated that the function g(z) is asymptotic in the rough sense of Poincaréto g(z) on the half-plane (z) > 0 at infinity. We will seee (cf. Com. 6.2.7) that the function g(z) is ℜ actually 1/2-Gevrey asymptotic to g(z). Our computations in Sect. 2.2.2 do not allowe us to state yet such a fact since we did not determine how the constant C depends on N. ⊲ The function h(z) of Examplee 2.2.7 was proved to be 1-Gevrey asymptotic to the series h(z) (cf. Estim. (10) on the right half-plane (z) > 0 at infinity. ℜ ⊲ The function ℓ(z) of Example 2.2.8 satisfies the same estimate (10) as h(z) on the sectore ′ = π/2+ δ < arg(z) < π/2 δ , for (0 <δ <π/2), with a constant C {− 1/2− } which can be chosen equal to C = e−1/2+e N!2N /(sin δ)N+1. The function ℓ(z) is then 1-Gevrey asymptotic to the series ℓ˜(z) on the right half-plane (z) > 0 at infinity. ℜ Proposition 2.3.10.— An s-Gevrey asymptotic series is an s-Gevrey series.

n Proof. — Suppose the series n≥0 anx is the s-Gevrey asymptotic series of a function f on . For all N, the result follows from Condition (12) applied P twice to N−1 N a xN = f(x) a xn f(x) a xn . N − n − − n n=0 n=0 X X

Proposition 2.3.11.— A function f ( ) belongs to s( ) if and only if ′ ′ ∈ A ′ A for all ⋐ there exist constants C ′ > 0 and A ′ > 0 such that the following estimate holds for all N N and x ′: ∈ ∈ N d f ′ s+1 ′N (13) (x) C ′ (N!) A ′ . dxN ≤

Proof.— Prove that Condition (13) implies Condition (12). — Like in the proof of Prop. 2.2.11, write Taylor’s formula with integral remainder:

N−1 f(x) a xn = x 1 (x t)N−1f (N)(t)dt = 1 xf (N)(t)d(x t)N − n 0 (N−1)! − − N! 0 − n=0 X R R and conclude that

N−1 N n 1 d f N ′ s ′N N f(x) anx sup N (t) x C ′ (N!) A ′ x . − ≤ N! t∈ ′ dx ·| | ≤ | | n=0 X Prove that Condition (12) implies Condition (13). — Like in the proof of Prop. 2.2.9, attach to any x ′ a circle γ centered at x with radius x δ, the ∈ x | | constant δ being chosen so small that γx be contained in and apply Cauchy’s 2.3. GEVREY ASYMPTOTICS 23 integral formula: N−1 dN f N! dt N! dt (x)= f(t) = f(t) a tn dxN 2πi (t x)N+1 2πi − n (t x)N+1 γx γx n=0 Z − Z X − since the N th derivative of a polynomial of degree N 1 is 0. Hence, − N N d f N! s N t ′ N (x) C (N!) A ′ | | N+1 dt dx ≤ 2π γ t x Z x | − | N N 1 s+1 N x (1 + δ) C ′ (N!) A ′ | | 2πδ x ≤ 2π x N+1δN+1 | | | | s+1 ′N ′ 1 = C ′ (N!) A ′ with A ′ = A ′ 1+ . δ Proposition 2.3.12 (Differential algebra and Taylor map) The set ( ) is a differential C-algebra and the Taylor map T restricted As to ( ) induces a morphism of differential algebras As T = T : ( ) C[[x]] s, As −→ s with values in the algebra of s-Gevrey series. Proof. — Let ′ ⋐ . Suppose f and g belong to ( ) and satisfy on ′ As dN f dN g (x) C(N!)s+1AN and (x) C′(N!)s+1A′N . dxN ≤ dxN ≤

The product fg belongs to ( ) (cf. Prop. 2.2.9) and its derivatives satisfy A N dN (fg) dpf dN−pg (x) Cp (x) (x) CC′(N!)s+1(A + A′)N . dxN ≤ N dxp dxN−p ≤ p=0 X The fact that the range T ( ) be included in C[[x]] follows from Propo- s, As s sition 2.3.10. Observe now the eﬀect of a change of variable x = tr, r N∗. Clearly, if ∈ a series f(x) is Gevrey of order s (level k) then the series f(tr) is Gevrey of order s/r (level kr). What about the asymptotics? Let e=]α,β[ ]0,R[ be a sector in (the directions α ande β are not given × 1/r modulo 2π) and let /r =]α/r, β/r[ ]0,R [ so that as the variable t runs r × over /r the variable x = t runs over . From Deﬁnition 2.3.7 we can state: Proposition 2.3.13 (Gevrey asymptotics in an extension of the variable) The following two assertions are equivalent: 24 CHAPTER 2. ASYMPTOTIC EXPANSIONS

(i) the function f(x) is s-Gevrey asymptotic to the series f(x) on ; (ii) the function g(t) = f(tr) is s/r-Gevrey asymptotic to g(t) = f(tr) e on /r. e e Way back, given an s′-Gevrey series g(t), the series f(x) = g(x1/r) exhibits, in general, fractional powers of x. To keep working with series of integer powers of x one may use rank reductione as followse [LR01e]. One can uniquely decompose the series g(t) as a sum

r−1 j r g(et)= t gj(t ) Xj=0 r e e r 2πi/r r where the terms gj(t ) are entire power series in t . Set ω = e and x = t . The series g (x) are given, for j = 0,...,r 1, by the relations j − e r−1 j r ℓ(r−j) ℓ e rt gj(t )= ω g(ω t). Xℓ=0 For j = 0,...,r 1, let j edenote the sector e − /r j =](α + 2jπ)/r, (β + 2jπ)/r[ ]0,R1/r[ /r × 0 j j r so that as t runs through /r = /r then ω t runs through /r and x = t runs through . From the previous relations and Proposition 2.3.13 we can state:

Corollary 2.3.14 (Gevrey asymptotics and rank reduction) The following two assertions are equivalent: (i) for ℓ = 0,...,r 1 the series g(t) is an s′-Gevrey asymptotic series − on ℓ/r (in the variable t); (ii) for j = 0,...,r 1 the r-ranke reduced series g (x) is an s′r-Gevrey − j asymptotic series on (in the variable x = tr). e With these results we might limit the study of Gevrey asymptotics to small values of s (s s ) or to large ones (s s ) at convenience. ≤ 0 ≥ 1 2.3.3. Flat s-Gevrey asymptotic functions. — In this section we address the following question: to characterize the functions that are both s- Gevrey asymptotic and ﬂat on a given sector . To this end, we introduce the notion of exponential ﬂatness. 2.3. GEVREY ASYMPTOTICS 25

Definition 2.3.15.— A function f is said to be exponentially flat of order k (or k-exponentially flat) on a sector if, for any proper subsector ′ ⋐ of , there exist constants K and A > 0 such that the following estimate holds for all x ′ : ∈ A (14) f(x) K exp ≤ − x k · | | The constants K and A may depend on ′.

Notation 2.3.16.— We denote the set of k-exponentially flat functions on by ≤−k( ). A Proposition 2.3.17.— Let be an open sector. The functions which are s-Gevrey asymptotically flat on are the k-exponentially flat functions, i.e., ( ) <0( )= ≤−k( ) (recall s = 1/k). As ∩ A A Proof.— ⊲ Let f ( ) <0( ) and prove that f ≤−k( ). ∈ As ∩ A ∈ A It is, here, more convenient to write Condition (12) in the following equivalent form: for all ′ ⋐ , there exist A> 0,C > 0 such that the estimate N f(x) CN N/k A x N = C exp ln N A x k ≤ | | k | | holds for all N and all x ′ (with possibly new constants Aand C). ∈ For x fixed, look for a lower bound of the right hand side of this estimate as N runs over N. The derivative ϕ′(N)=ln N(A x )k + 1 of the function | | ϕ(N)= N ln N(Ax )k | | seen as a function of a real variable N > 0 vanishes at N = 1/ e(A x k) 0 | | and ϕ reaches its minimal value ϕ(N ) = N at that point. Taking into 0 − 0 account the monotonicity of ϕ, for instance to the right of N0, we can assert that 1 1 inf ϕ(N) ϕ(N0 + 1) = ϕ(N0) 1+ 1 (1 + N0)ln 1+ . N∈N ≤ N0 − N0 Substituting this value of N0 as a function of x in ϕ, we can write ϕ(N0 + 1) = ϕ(N0)ψ(x) where ψ(x) is a bounded function on . Hence, there exists a constant C′ > 0 such that f(x) C′ exp a with a = 1 > 0 independent of x ′. | | ≤ − |x|k k eAk ∈ This proves that f belongs to ≤−k( ). A 26 CHAPTER 2. ASYMPTOTIC EXPANSIONS

⊲ Let f ≤−k( ) and prove that f ( ) <0( ). ∈ A ∈ As ∩ A The hypothesis is now: for all ′ ⋐ , there exist A > 0,C > 0 such that an estimate A f(x) C exp ≤ − x k | | holds for all x ′. Hence, for any N, the estimate ∈ A f(x) x −N C exp x −N . ·| | ≤ − x k | | | | For N ﬁxed, look for an upper bound of the right hand side of this estimate as x runs over R+. Let ψ( x ) = exp A x −N . Its logarithmic derivative | | | | − |x|k | | ψ′( x ) N Ak | | = + ψ( x ) − x x k+1 | | | | | | vanishes for Ak/ x k = N and ψ reaches its maximum value at that point. | | Thus, max ψ( x ) = exp N N N/k and there exists constants |x|>0 | | − k Ak a =(eAk)−1/k and C > 0 such that, for all N N and x Σ′ the function f ∈ ∈ satisﬁes f(x) CN N/k a x N . ≤ | | <0 Hence, f belongs to s( ) ( ). A ∩ A

2.4. The Borel-Ritt Theorem With any asymptotic function f ( ) over a sector the Taylor map T ∈ A associates a formal series f = T (f). We address now the converse problem: is any formal series the Taylor series of an asymptotic function over a given sector ? The theorem belowe states that the answer is yes for any open sector with ﬁnite radius in C∗ or in Poincar´easymptotics. In case the series is s-Gevrey an s-Gevrey asymptotic function always exists when the opening of the sector is small enough but we will see on examples that it might not exist for a too wide . Notice that the Taylor series of a function f (C∗) ∈ A is necessarily convergent by the removable singularity Theorem of Riemann. And thus, when is included in C∗, it cannot be a full neighborhood of 0 in C∗.

Theorem 2.4.1 (Borel-Ritt).— Let = C∗ be an open sector of C∗ or of 6 the Riemann surface of logarithm with ﬁnite radius R. (i) (Poincar´easymptotics) The Taylor map T : ( ) C[[x]] is onto. A → 2.4. THE BOREL-RITT THEOREM 27

(ii) (Gevrey asymptotics) Suppose has opening π/k. Then, the | | ≤ Taylor map T : ( ) C[[x]] is onto. Recall s = 1/k. s, As → s Proof. — (i) Poincar´easymptotics. — Various proofs exist. The one presented here can be found in [Mal95]. For simplicity, begin with the case of a sector in C∗. ⊲ Case when lies in C∗. Modulo rotation it is suﬃcient to consider the case when = −π,+π(R) is the disc of radius R slit on the real negative axis.

Figure 6

Given any series a xn C[[x]] we look for a function f ( ) n≥0 n ∈ ∈ A with Taylor series T f = a xn. To this end, one introduces functions P n≥0 n β (x) ( ) satisfying the two conditions n ∈O P (1) : a β (x)xn ( ) and (2) : T β (x) 1 for all n 0. n n ∈O n ≡ ≥ nX≥0 Such functions exist: consider, for instance, the functions β 1 and, for 0 ≡ n 1, β (x) = 1 exp b /√x with positive b and √x the principal ≥ n − − n n determination of the square root. In view to Condition (1), observe that since 1 ez = z et dt − − 0 then 1 ez < z for (z) < 0. This implies β (x) b / x for all | − | | | ℜ | n | ≤ n | |R x and n 1 and then, ∈ ≥ p a β (x)xn a b x n−1/2 a b Rn−1/2. n n ≤ | n| n | | ≤ | n| n Now, choose b such that the series a b Rn−1/2 be convergent. n n≥1 | n| n Then, the series a β (x)xn converges normally on and its sum f(x)= n≥0 n n P a β (x)xn is holomorphic on . n≥0 n n P To prove Condition (2), consider any proper sub-sector ′ ⋐ of and P x ′. Then, for any N 1, we can write ∈ ≥ N−1 N−1 f(x) a xn a (β (x) 1)xn + x N a β (x)xn−N . − n ≤ n n − | | n n n=0 n=0 n≥N X X X The first summand is a finite sum of terms all asymptotic to 0 and then, is majorized by C′ x N , for a convenient positive constant C′. The second | | 28 CHAPTER 2. ASYMPTOTIC EXPANSIONS summand is majorized by x N 2 a + a b Rn−1/2−N . | | | N | | n| n n≥XN+1 Choosing C = C′ + 2 a + a b Rn−1/2N provides a positive con- | N | n≥N+1 | n| n stant C (independent of x but depending on N and ′) such that P N f(x) a xn C x N for all x ′. − n ≤ | | ∈ n=0 X This ends the proof in this case. ⊲ General case when lies in C∗. — It is again sufficient to consider the case of a sector of the form = x C∗ ; arg(x) < kπ, 0 < x

Let f(x) C[[x]]s be an s-Gevrey series which, up to a polynomial, we ∈ n may assume to be of the form f(x) = n≥k anx . It is suﬃcient to consider a sector eof opening π/k (as always, k = 1/s) and by means of a rotation, we P can then assume that is an opene sector bisected by the direction θ = 0 with opening π/k; we denote by R its radius. We must ﬁnd a function f ( ), ∈ As s-Gevrey asymptotic to f over . The proof used here is based on the Borel and the Laplace transforms which will be at the coree of Borel-Laplace summation in Section 6.3. Since f(x) is an s-Gevrey series (cf. Def. 2.3.1) its k-Borel transform(1) a f(ξ)= n ξn−k e Γ(n/k) nX≥k is a convergent series(2) andb we denote by ϕ(ξ) its sum. The adequate Laplace transform to “invert” the k-Borel transform (as a function ϕ(ξ), not as a series f(ξ)) in the direction θ = 0 would be the k-Laplace transform +∞ −ζ/xk b k(ϕ)(x)= φ(ζ)e dζ L Z0

(1) n See Sect. 6.3.1. The k-Borel transform of a series Pn≥k anx is the usual Borel transform of the n/k k s series Pn≥k anX with respect to the variable X = x and expressed in the variable ξ = ζ . (2) Although, when k is not an integer, the series f(ξ) is not a series in integer powers of ξ it becomes b so after factoring by ξ−k. We mean here that the power series ξkf(ξ) is convergent. b 2.4. THE BOREL-RITT THEOREM 29 where ζ = ξk and φ(ζ) = ϕ(ζ1/k). However, although the series f(ξ) is convergent, its sum ϕ(ξ) cannot be analytically continued along R+ up to inﬁnity in general. So, we choose b> 0 belonging to the disc of convergenceb of f(ξ) and we consider a truncated k-Laplace transform

bk b k (15) f b(x)= φ(ζ)e−ζ/x dζ Z0 instead of the full Laplace transform (ϕ)(x). Lemma 2.4.2 below shows that Lk the function f = f b answers the question.

Lemma 2.4.2 (Truncated Laplace transform).— With notations and conditions as above, and especially being an open sector bisected by θ = 0 with opening π/k, the truncated k-Laplace transform f b(x) of the sum ϕ(ξ) of the k-Borel transform of f(x) in direction θ = 0 is s-Gevrey asymptotic to f(x) on (with s = 1/k as usually). e Proofe . — Given 0 < δ < π/2 and R′ < R, consider the proper sub-sector of deﬁned by = x ; arg(x) < π/(2k) δ/k and x 0 then, ζ(n/k)−1 e−ζ/x bn−k for all ζ [0,bk]. Consequently, ℜ k ≤ ∈ an (n/k)−1 −ζ/x k the series n≥k Γ(n/k) ζ e converges normally on [0,b ] and we can permute sum and integral. Hence, P N−1 bk b n an (n/k)−1 −ζ/xk f (x) anx = ζ e dζ − Γ(n/k) 0 nX=k nX≥N Z N−1 +∞ a k n ζ(n/k)−1 e−ζ/x dζ. − Γ(n/k) bk nX=k Z (n/k)−1 (N/k)−1 However, ζ/bk ζ/bk both when ζ bk and n N ≤ | | ≤ ≥ and when ζ bk and n

| | ≥ 30 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Comment 2.4.3 (On the Euler series (Exa. 2.2.4))

The proof of the Borel-Ritt Theorem provides inﬁnitely many functions asymptotic to the Euler series E(x)= ( 1)nn! xn+1 at 0 on the sector = x ; arg(x) < 3π/2 . n≥0 − { | | } For instance, the following family provides inﬁnitely many such functions: P e n −a/((n!)2x1/3) n+1 Fa(x)= ( 1) n! 1 e x , a> 0. − − n≥0 X − +∞ e ξ/x We saw in Example 2.2.4 that the Euler function E(x)= 0 1+ξ dξ is both solution of the Euler equation and asymptotic to the Euler series on . We claim that it is the unique R function with these properties. Indeed, suppose E1 be another such function. Then, the

difference E(x) E1(x) would be both asymptotic to the null series 0 on and solution − of the homogeneous associated equation x2y′ + y = 0. However, the equation x2y′ + y =0

admits no such solution on but 0. Hence, E = E1 and the inﬁnitely many functions given by the proof of the Borel-Ritt Theorem do not satisfy the Euler equation in general.

Taking into account Props. 2.2.9, 2.3.12 and 2.3.17 we can reformulate the Borel-Ritt Theorem 2.4.1 as follows.

Corollary 2.4.4.— The set <0( ) of flat functions on and the set A ≤−k( ) of k-exponentially flat functions on are differential ideals of ( ) A A 2.5. THE CAUCHY-HEINE THEOREM 31 and ( ) respectively. The sequences As T 0 <0( ) ( ) C[[x]] 0 → A −→ A −−→ → and, when π/k, | | ≤ T 0 ≤−k( ) ( ) s, C[[x]] 0 → A −→ As −−−→ s → are exact sequences of morphisms of differential algebras. The Borel-Ritt Theorem implies the classical Borel Theorem in the real case providing thus a new proof of it. Corollary 2.4.5 (Classical Borel Theorem).— Any formal power series a xn C[[x]] is the Taylor series at 0 of n≥0 n ∈ a ∞-function of a real variable x. C P Proof. — Apply the Borel-Ritt Theorem on a sector ′ containing R+ and on a sector ′′ containing R−. The two functions so obtained glue together at 0 into a ∞-function in a neighborhood of 0 in R. C

2.5. The Cauchy-Heine Theorem In this section we are given: • ∗ ⊲ a sector = α,β(R) with vertex 0 in C ; • • ⊲ a point x0 in and the straight path γ = ]0,x0] in ; • • ⊲ a function ϕ <0( ) flat at 0 on . ∈ A Definition 2.5.1.— One defines the Cauchy-Heine integral associated with ϕ and x0, to be the function 1 ϕ(t) f(x)= dt. 2πi t x Zγ −

Figure 7 32 CHAPTER 2. ASYMPTOTIC EXPANSIONS

Denote by:

⊲ = α,β+2π(R) a sector with vertex 0 in the Riemann surface of loga- • rithm overlapping on ;

⊲ θ0 the argument of x0 satisfying α<θ0 <β;

⊲ γ = θ0,θ0+2π( x0 ) the disc of radius x0 slit along γ; D• • | | | | ⊲ ′ = x < x = ( x ); ∩{| | | 0|} α,β | 0| ⊲ ′ = x < x = ( x ). ∩{| | | 0|} α,β+2π | 0| The Cauchy-Heine integral determines a well-defined and analytic function f on γ. By Cauchy’s Theorem, Cauchy-Heine integrals associated with D • 1 ϕ(t) different points x0 and x1 in differ by ⌢ dt, an analytic function 2πi x0x1 t−x on a neighborhood of 0. R Theorem 2.5.2 (Cauchy-Heine).— With notations and conditions as be- • 1 ϕ(t) fore and especially, ϕ flat on , the Cauchy-Heine integral f(x)= 2πi γ t−x dt has the following properties: R 1. The function f can be analytically continued from to ′; we also use Dγ the term Cauchy-Heine integral when referring to this analytic continuation which we keep denoting by f. 2. The function f belongs to ( ′). A 3. Its Taylor series at 0 on ′ reads

n 1 ϕ(t) ′ T f(x)= anx with an = n+1 dt. 2πi γ t nX≥0 Z • 4. Its variation varf(x)= f(x) f(xe2πi) is equal to ϕ(x) for all x ′. −≤−k • ′ ∈ 5. If, in addition, ϕ belongs to ( ) then, f belongs to s( ) with the A • A above Taylor series, i.e., , if ϕ is k-exponentially flat on then, f is s-Gevrey n ′ asymptotic to the above series n≥0 anx on (recall s = 1/k). Proof. — The five steps can beP proved as follows. 1. — Consider, for instance, the function f for values of x on the left of γ. To analytically continue this “branch” of the function f to the right of γ it suffices to deform the path γ by pushing it to the right keeping its endpoints ′ 0 and x0 fixed. This allows us to go up to the boundary arg(x)= α of . We can similarly continue the “branch” of the function f defined for values of x on the right of γ up to the boundary arg(x)= β + 2π of ′. 2–3. — We have to prove that, for all subsector ′′ ⋐ ′, the function f satisfies the asymptotic estimates of Definition 2.2.1. 2.5. THE CAUCHY-HEINE THEOREM 33

⊲ Suppose ﬁrst that ′′ γ = . Writing ∩ ∅ N−1 1 xn xN = + t x tn+1 tN (t x) n=0 − X − as in Example 2.2.4, we get

N−1 xN ϕ(t) f(x)= a xn + dt. n 2πi tN (t x) n=0 γ X Z −

Figure 8

Given x ′′, then t x dist(t, ′′)= t sin(δ) for all t γ and so ∈ | − | ≥ | | ∈ N−1 (16) f(x) a xn C x N − n ≤ | | n=0 X where the constant C = 1 |ϕ(t)| dt is finite (the integral converges 2π γ |t|N+1 sin(δ) ′′ since ϕ is flat at 0 on γ) and R depends on N and , but is independent of x ′′. ∈ ⊲ Suppose now that ′′ γ = . Push homotopically γ into a path made ∩ 6 ∅ of the union of a segment γ1 = ]0,x1] and a curve γ2, say a circular arc, joining ′′ x1 to x0 without meeting as shown on the figure. The integral splits into two parts f1(x) and f2(x).

Figure 9 34 CHAPTER 2. ASYMPTOTIC EXPANSIONS

The term f1(x) belongs to the previous case and is then asymptotic to

1 ϕ(t) n ′′ n+1 dt x on . 2πi γ1 t nX≥0 Z The term f2(x) deﬁnes an analytic function on the disc x < x0 and is 1 ϕ(t) n | | | | asymptotic to its Taylor series n+1 dt x . Hence, the result. n≥0 2πi γ2 t • • 4. — Given x ′ compute the variation of f at x. Recall that x ′ ∈ P R ∈ means that x belongs to the ﬁrst sheet of ′. So, as explained in the proof of point 1, to evaluate f(x) we might have to push homotopically the path γ to the right into a path γ′. When x lies to the left of γ we can keep γ′ = γ. To evaluate f(xe2πi) we might have to push homotopically the path γ to the left into a path γ′′ taking γ′′ = γ when x lies to the right of γ. • The concatenation of γ′ and γ′′ generates a path Γ in enclosing x and − • since the function ϕ(t)/(t x) is meromorphic on we obtain by the Cauchy’s − Residue Theorem: 1 ϕ(t) ϕ(t) (17)var f(x)= f(x) f(xe2πi)= dt = Res ,t = x = ϕ(x) − 2πi Γ t x t x Z − −

Figure 10

5. — Given ′′ ⋐ ′ suppose that the function ϕ satisﬁes ϕ(x) K exp A/ x k on ′′. ≤ − | | Consider the case when ′′ γ = . Then, the constant C in estimate (16) ∩ ∅ satisﬁes k K exp( A/ t ) ′ −N/k C −N+1| | dt C A Γ(N/k) ≤ 2π γ t ≤ Z | | with a constant C′ > 0 independent of N. The case when ′′ γ = is treated ∩ 6 ∅ similarly by deforming the path γ as in points 2–3. Hence, f(x) is s-Gevrey n ′ asymptotic to the series n≥0 anx on . P 2.5. THE CAUCHY-HEINE THEOREM 35

Comments 2.5.3 (On the Euler function (Exa. 2.2.4))

• Set = α,β ( ) with α = 3π/2 and β = π/2 and = α,β+2π( ). Let E(x) denote ∞ − − ∞ the Euler function as in Example 2.2.4 and, given θ, let dθ denote the half line issuing from 0 with direction θ. • ⊲ The variation of the Euler function E on is given by e−ξ/x e−ξ/x var E(x) = dξ dξ (ε small enough) 1+ ξ − 1+ ξ Zd−π+ε Zdπ−ε e−ξ/x = 2πi Res ,ξ = 1 (Cauchy’s Residue Thm.) − 1+ ξ − = 2πie1/x. − ⊲ Apply the Cauchy-Heine Theorem by choosing the 1-exponentially flat func- • • 1/x tion ϕ(x)= 2πie on and a point x0 , for instance x0 = r real negative. • ′ − ′ ′ ∈ − Denote = x < x0 and = x < x0 . The Cauchy-Heine Theorem ∩{| | | |} ∩ {| | | |} ′ provides a function f which, as the Euler function, belongs to 1( ) with variation ϕ(x) • A on ′. We claim that E and f differ by an analytic function near 0. Indeed, the Taylor series ′ n of f on reads n≥0 an,x0 x with coefficients

x0 1/t +∞ P e n−1 n−1 −u an,x = dt =( 1) u e du 0 − tn+1 − Z0 Z1/r while the Taylor coeﬃcients an of the Euler function E are given by a0 = 0 and for n 1 ≥ by an = limr→+∞ an,x . Since a0,x has no limit as r tends to + we consider, instead 0 0 ∞ of f, the function −r +∞ −u 1 1 1/t xe f(x) a0,x0 = e dt = du. − − 0 t x − t 1/r 1+ ux Z − Z Suppose x = x eiθ. Then, the Euler function at x can be deﬁned by the integral | | e−ξ/x +∞ xe−u E(x)= dξ = du 1+ ξ 1+ xu Zdθ Z0 and 1/r e−u E(x) f(x)= a0,x + x du − − 0 1+ ux Z0 which is an analytic function on the disc x

deduced from the Euler equation (1) by dividing it by x and then, differentiating once. Since the equation has no singular point but 0 (and infinity) the Cauchy-Lipschitz Theorem allows one to analytically continue the Euler function along any path which avoids 0 and then in particular, outside of the sector 3π/2 < arg(x) < +3π/2. However, when − crossing the lateral boundaries of this sector the Euler function E(x) stops being asymptotic to the Euler series at 0; it even stops having an asymptotic expansion since, from the variation formula above (cf. also the end of Exa. 2.2.4), one has now to take into account an exponential term which is unbounded. This phenomenon is known under the name of Stokes phenomenon. It is at the core of the meromorphic classification of linear differential equations (cf. Sect. 4.3).

Exercise 2.5.4. — Study the asymptotics at 0 of the function +∞ e−ξ/x F (x)= dξ. ξ2 + 3ξ + 2 Z0 and its analytic continuation. Compute its variation. CHAPTER 3

SHEAVES AND CECHˇ COHOMOLOGY WITH AN INSIGHT INTO ASYMPTOTICS

In this chapter, we recall some deﬁnitions and results used later and some examples, about sheaves and Cechˇ cohomology. For more precisions we refer to the classical literature (cf. [God58], [Ten75], [Ive86] for instance).

3.1. Presheaves and sheaves Sheaves are the adequate tool to handle objects defined by local conditions without having to make explicit how large is the domain of validity of the conditions. They are mainly used as a bridge from local to global properties. It is convenient to start with the weaker concept of presheaves which we usually denote with an overline. 3.1.1. Presheaves. — Let us start with the definition of presheaves with values in the category of sets and continue with the case of various sub- categories (for the definition of a category, see for instance [God58, Sect. 1.7]). Definition 3.1.1 (Presheaf).— A presheaf (of sets) over a topological F space X called the base space is defined by the following data: (i) to any open set U of X there is a set (U) whose elements are called F sections of on U; F (ii) to any couple of open sets V U there is a map ρ : (U) (V ) ⊆ V,U F → F called restriction map satisfying the two conditions:

⊲ ρU,U = idU for all U, ⊲ ρ ρ = ρ for all open sets W V U. W,V ◦ V,U W,U ⊆ ⊆ In the language of categories, a presheaf of sets over X is then a contravariant functor from the category of open subsets of X into the category of sets. 38 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

Unless otherwise specified, we assume that X does not reduce to one element. The names “section” and “restriction map” take their origin in Example 3.1.2 below which, with the notion of espace étalé (cf. Def. 3.1.10), will become a reference example.

Example 3.1.2 (A fundamental example).— Let F be a topological space and π : F X a continuous map. A presheaf is associated with F and π as follows: → F for all open set U in X one deﬁnes (U) as the set of sections of π on U, i.e., continuous F maps s : U F such that π s = idU . The restriction maps ρV,U for V U are deﬁned → ◦ ⊆ by ρV,U (s)= s|V .

Example 3.1.3 (Constant presheaf).— Given any set (or group, vector space, etc...) C, the constant presheaf X over X is deﬁned by X (U) = C for all C C open set U in X and the maps ρV,U = idC : C C as restriction maps. →

Example 3.1.4 (An exotic example).— Given any marked set with more than one element, say (X = C, 0), one deﬁnes a presheaf over X as follows: (X)= X G G and (U) = 0 when U = X; all the restriction maps are equal to the null maps except G { } 6 ρX,X which is the identity on X.

Below, we consider presheaves with values in a category equipped with C an algebraic structure. We assume moreover that, in , there exist products, C the terminal objects are the singletons, the isomorphisms are the bijective morphisms. The same conditions will apply to the sheaves we consider later on.

Definition 3.1.5.— A presheaf over X with values in a category is a C presheaf of sets satisfying the following two conditions: (iii) For all open set U of X the set (U) is an object of the category ; F C (iv) For any couple of open sets V U the map ρ is a morphism in . ⊆ V,U C In the next chapters, we will mostly be dealing with presheaves or sheaves of modules, in particular, of Abelian groups or vector spaces, and presheaves or sheaves of differential C-algebras, i.e., presheaves or sheaves with values in a category of modules, Abelian groups, or vector spaces and presheaves or sheaves with values in the category of differential C-algebras. 3.1. PRESHEAVES AND SHEAVES 39

Deﬁnition 3.1.6 (Morphism of presheaves).— Given and two F G presheaves over X with values in a category , a morphism f : is a C F → G collection, for all open sets U of X, of morphisms f(U): (U) (U) F −→ G in the category which are compatible with the restriction maps, i.e., such that C the diagrams f(U) (U) (U) F −−−−→ G ′ ρV,U ρV,U  f(V )  (V) (V ) F y −−−−→ Gy commute (ρ and ρ′ denote the restriction maps in and respectively). V,U V,U F G Deﬁnition 3.1.7.— A morphism f of presheaves is said to be injective or surjective when all morphisms f(U) are injective or surjective.

The morphisms of presheaves from into form a set, precisely, they F G form a subset of Hom (U), (U) . Composition of morphisms in the U⊆X F G category induces composition of morphisms of presheaves over X with values C Q in . It follows that presheaves over X with values in form themselves a C C category. When is Abelian, the category of presheaves over X with values in is C C also Abelian. In particular, one can talk of an exact sequence of presheaves

fj fj+1 · · · → F j−1 −→ F j −−→ F j+1 →··· which means that the following sequence is exact for all open set U:

f (U) f (U) (U) j (U) j+1 (U) . · · · → F j−1 −−→ F j −−−→ F j+1 →··· The category of modules over a given ring, hence also the category of Abelian groups and the category of vector spaces, are Abelian. They admit the trivial module 0 as terminal object. { } The category of rings, and in particular, the category of diﬀerential C- algebras, is not Abelian. Although the quotient of a ring by a subring A J is not a ring in general, this becomes true when is an ideal and allows one J to consider short exact sequences 0 / 0 of presheaves of → J → A → A J → rings or of diﬀerential C-algebras. 40 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

Deﬁnition 3.1.8 (Stalk).— Given a presheaf over X and x X, the F ∈ stalk of at x is the direct limit F = lim (U), x −→ F U∋x F the limit being taken on the ﬁltrant set of the open neighborhoods of x in X ordered by inclusion. The elements of are called germs of sections of F x F at x.

Let us first recall what is understood by the terms direct limit and filtrant. ⊲ The direct limit E = lim(E ,f ) −→ α β,α α∈I of a direct family (E ,f : E E for α β)(i.e., it is required that the α β,α α −→ β ≤ set of indices I be ordered and right filtrant which means that given α,β I ∈ there exists γ I greater than both α and β; moreover, the morphisms must ∈ satisfy f = id and f f = f for all α β γ) is the quotient α,α α γ,β ◦ β,α γ,α ≤ ≤ of the sum F = E of the spaces E by the equivalence relation : for α∈I α α R x E and y E , one says that ∈ α ∈ Fβ x y if there exists γ such that γ α, γ β and f (x)= f (y). R ≥ ≥ γ,α γ,β

In the case of a stalk here considered, the maps fβ,α are the restriction maps ρV,U . ⊲ Filtrant means here that, given any two neighborhoods of x, there exists a neighorhood smaller than both of them. Their intersection, for example, provides such a smaller neighborhood.

Thus, a germ ϕ at x is an equivalence class of sections under the equivalence relation: given two open sets U and V of X containing x, two sections s (U) and t (V ) are equivalent if and only if there is an open set ∈ F ∈ F W U V containing x such that ρ (s)= ρ (t). ⊆ ∩ W,U W,V By abuse and for simplicity, we allow us to say “the germ ϕ at x” when ϕ is an element of (U) with U x identifying so the element ϕ in the equivalence F ∋ class to the equivalence class itself. Given s (U) and t (V ) one should be aware of the fact that the ∈ F ∈ F equality of the germs s = t for all x U V does not imply the equality of x x ∈ ∩ the sections themselves on U V . ∩ 3.1. PRESHEAVES AND SHEAVES 41

A counter-example is given by taking the sections s 0 and t 1 whose germs are ≡ ≡ everywhere 0 in Example 3.1.4.

Also, it is worth to notice that a consistent collection of germs for all x U ∈ does not imply the existence of a section s (U) inducing the given germs ∈ F at each x U. Consistent means here that any section v (V ) representing ∈ ∈ F a given germ at x induces the neighboring germs: there exists an open sub- neighborhood V ′ V U of x where the given germs are all represented by ⊆ ⊆ v.

A counter-example is given by the constant presheaf X when X is disconnected. Consider, C ∗ for instance, X = R ,C = R and the collection of germs sx = 0 for x < 0 and sx = 1 for x> 0. The presheaf deﬁned in Section 3.1.5 will provide another example. A

Such inconveniences are circumvented by restricting the notion of presheaf to the stronger notion of sheaf given just below.

3.1.2. Sheaves. —

Deﬁnition 3.1.9 (Sheaf).— A presheaf over X is a sheaf (we denote it F then by ) if, for all open set U of X, the following two properties hold: F 1. If two sections s and σ of (U) agree on an open covering = U F U { j}j∈J of U (i.e., if they satisfy ρUj ,U (s)= ρUj ,U (σ) for all j) then s = σ. 2. Given any consistent family of sections s (U ) on an open cov- j ∈ F j ering = U of U there exists a section s (U) gluing all the s ’s U { j}j∈J ∈ F j (i.e., such that for all j, ρUj ,U (s)= sj). Consistent means here that, for all i,j, the restrictions of si and sj agree on U U , i.e., ρ (s )= ρ (s ). i ∩ j Ui∩Uj ,Ui i Ui∩Uj ,Uj j The presheaf of Example 3.1.2 is a sheaf. In Example 3.1.4 Condition 1 fails. In the F case of the constant presheaf over a disconnected base space X (cf. Exa. 3.1.3) and in the case of the presheaf in the next section Condition 2 fails. A It follows from the axioms of sheaves that ( ) is a terminal object. F ∅ Thus, ( )= 0 when is a sheaf of modules, Abelian groups and vector F ∅ { } F spaces or of diﬀerential C-algebras. When is a sheaf of modules the restriction maps are linear and Condi- F tion 1 reduces to: a section which is zero in restriction to a covering = U U { j} is the null section. 42 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

3.1.3. From presheaves to sheaves: espaces étalés. — With any presheaf there is a sheaf canonically associated as follows. Consider the F F space F = (disjoint union of the stalks of ) and endow it with the x∈X F x F following topology: a set Ω F is open in F if, for all open set U of X and F ⊆ all section s (U), the set of all elements x U such that the germ s of s ∈ F ∈ x at x belong to Ω is open in X. Given s (U) where U is an open subset of X, consider the map ∈ F s˜ : U F defined bys ˜(x)= s . Denote by π the projection map π : −→ x F X, s π(s ) = x. The topology on F is the less fine for whichs ˜ is → x 7→ x continuous for all U and s, and the topology induced on the stalks = π−1(x) F x is the discrete topology. The setss ˜(U) are open in F and the mapss ˜ satisfy π(˜s(x)) = x for all x U. It follows that π is a local homeomorphism. ∈ Definition 3.1.10 (Espace étalé, associated sheaf)

⊲ The topological space F is called the espace étalé over X associated with . F ⊲ The sheaf associated with the presheaf is the sheaf of continuous F F sections of π : F X as defined in Example 3.1.2. →

Example 3.1.11 (Constant sheaf).— The espace ´etal´eassociated with the constant presheaf X in Example 3.1.3 is the topological space X C endowed with the C × topology product of the given topology on X and of the discrete topology on C. Whereas

the sections of X are the constant functions over X, the sections of the associated sheaf C X are all locally constant functions. The sheaf X is commonly called the constant sheaf C C over X with stalk C. Since there is no possible confusion one calls it too, the constant sheaf C over X using the same notation for the sheaf and its stalks.

The maps i(U) given, for all open subsets U of X, by i(U): (U) (U), s s˜ F −→ F 7−→ define a morphism i of presheaves. These maps may be neither injective (failure of condition 1 in Def. 3.1.9. See Exa. 3.1.4) nor surjective (failure of Condition 2 in Def. 3.1.9. See Exa. 3.1.11 or 3.1.22). One can check that the morphism i is injective when Condition 1 of sheaves (cf. Def. 3.1.9) is satisfied and that it is surjective when both Conditions 1 and 2 are satisfied, and so, we can state Proposition 3.1.12.— The morphism i is an isomorphism of presheaves if and only if is a sheaf. F 3.1. PRESHEAVES AND SHEAVES 43

In all cases, i induces an isomorphism between the stalks and at F x Fx any point x X. ∈ The morphism of presheaves i satisﬁes the following universal property: Suppose is a sheaf; then, any morphism of presheaves ψ : can G F →G be factored uniquely through the sheaf associated with , i.e., there exists F F a unique morphism ψ such that the following diagram commutes:

From the fact that,when is itself a sheaf, the morphism i is an isomor- F phism of presheaves between any presheaf and its associated sheaf, one can always think of a sheaf as being the sheaf of the sections of an espace étalé F π X. From that viewpoint, it makes sense to consider sections over any −→ subset W of X, open or not, and also to define any section as a collection of germs. Not any collection of germs is allowed. Indeed, if ϕ (W ) repre- ∈ F x sents the germ sx on a neighborhood Wx of x then, for the section s : W F ′ → to be continuous at x, the germs sx′ for x close to x must also be represented by ϕ. The set (W ) of the sections of a sheaf over a subset W of X is F F widely denoted by Γ(W ; ). F Recall the following definition (see end of Sect. 3.1.1 and Def. 3.1.9).

Deﬁnition 3.1.13 (Consistency).— ⊲ A family of sections s (W ) is said to be consistent if, when W W j ∈ F j i ∩ j is not empty, the restrictions of s and s to W W coincide. i j i ∩ j ⊲ A family of germs is said to be consistent if any germ generates its neighbors.

One can state:

Proposition 3.1.14.— Given a sheaf over X and W any subset of X, F open or not, a family of germs (s ) is a section of over W if and only x x∈W F if it is consistent.

Deﬁnition 3.1.15.— Let be the sheaf associated with a presheaf . We F F deﬁne a local section of to be any section of the presheaf . F F 44 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

Considering representatives of the germs s of a section s Γ(W ; ), x ∈ F Proposition 3.1.14 can be reformulated as follows. Proposition 3.1.16.— Let be the sheaf associated with a presheaf F F over X and let W be any subset of X, open or not. Sections of over W can F be seen as consistent collections of local sections s (U ) with U open in j ∈ F j j X and W U . ⊆ j j Clearly, suchS collections are not unique. When W is not open the inclusion W U is proper and the section ⊆ j j lives actually on a larger open set (the size of which depends not only on W S but both on W and the section).

3.1.4. Morphisms of sheaves. — Definition 3.1.17 (Sheaf morphism).— A morphism of sheaves is just a morphism of presheaves. With this definition, Proposition 3.1.12 has the following corollary. Corollary 3.1.18.— Let be a sheaf and ′ its associated sheaf when con- F F sidered as a presheaf. Then, and ′ are isomorphic sheaves. F F ′ Given two sheaves and ′ over X, let F π X and F ′ π X′ be their F F −→ −→ respective espace étalé. From the identification of a sheaf to its espace étaléa morphism f : ′ of sheaves can be seen as a continuous map, which can F → F also be denoted safely by f, between the associated espaces étalés with the condition that the following diagram commute:

Like presheaves, sheaves with values in a given category and their mor- C phisms form a category which is Abelian when is also Abelian. The category C of sheaves and the category of espaces étalés with values in a given category are equivalent. C Definition 3.1.19.— A morphism f : ′ of sheaves over X is said to F → F be injective (resp. surjective, resp. an isomorphism) if, for any x X, the ∈ stalk map f : ′ is injective (resp. surjective, resp. bijective). x Fx → Fx 3.1. PRESHEAVES AND SHEAVES 45

When a morphism f : ′ is injective then, for all open subset U of F → F X, the map f(U) : (U) ′(U) is injective. However, the fact that f be F → F surjective does not imply the surjectivity of the maps f(U) for all U; hence, a surjective morphism of sheaves is not necessarily surjective as a morphism of presheaves, the converse being, of course, true since the functor direct limit is exact.

Example 3.1.20.— Take for the sheaf of germs of holomorphic functions on F X = C∗ and for ′ the subsheaf (see Def. 3.1.21 below) of the non-vanishing functions. F The map f : ϕ exp ϕ is a morphism from to ′ which is surjective as a morphism 7→ ◦ F F of sheaves since the logarithm exists locally on C∗. However, the logarithm is not deﬁned as a univaluate function on all of C∗ and so, the map f is not a surjective morphism of presheaves. For instance, the identical function Id : x x cannot be written in the 7→ form Id = f(ϕ) for any ϕ in (C∗) or more generally, any ϕ in (U) as soon as U is not F F simply connected in C∗.

Deﬁnition 3.1.21.— A sheaf over X is a subsheaf of a sheaf over X F G if, for all open set U, it satisﬁes the conditions ⊲ (U) (U), F ⊆G .the inclusion map (U) ֒ (U) commute to the restriction maps ⊳ F →G .The inclusion j : ֒ is an injective morphism of sheaves F →G

3.1.5. Sheaves of asymptotic and of s-Gevrey asymptotic func- A As tions over S1.— The sheaves and of asymptotic functions we intro- A As duce in this section play a fundamental role in what follows. ⊲ Topology of the base space S1. — The base space S1 is the circle of directions from 0. One should consider it as the boundary of the real blow up of 0 in C, i.e., as the boundary S1 0 of the space of polar coordinates × { } (θ,r) S1 [0, [. ∈ × ∞ For simplicity, we denote S1 for S1 0 . × { } The map π : C deﬁned by π(θ,r) = r eiθ sends S1 to 0 → and S1 homeomorphically to C∗. A basis of open sets of S1 is \ given by the arcs I = ]θ0,θ1[ seen as the direct limit of the domains ˇ = I ]0,R[ in as R tends to 0. Such domains are identiﬁed via π to × sectors = x = r eiθ; θ <θ<θ and 0

Figure 1

I are given by (I)= lim ( ). −→ I,R A R→0 A Suppose an element of (I) is represented by two functions ϕ ( ) A ∈ A I,R and ψ ( ) on the same sector . This means that there exists a sub- ∈ A I,R I,R sector I,R′ of I,R on which ϕ and ψ coincide. By analytic continuation, we conclude that ϕ = ψ on all of I,R. Choosing as restriction maps the usual restriction of functions, this deﬁnes a presheaf of diﬀerential C-algebras. The example below shows that such a presheaf is not a sheaf.

Example 3.1.22.— Consider the lacunar series (see [Rud87, Hadamard’s Thm. 16.6 and Exa. 16.7]) 2n n/2 f1(x)= an(x 1) with an = exp( 2 ). − − nX≥0 − 2 n Since limsup an = 1 its radius of convergence as a series in powers of x 1 is n→+∞ | | − equal to 1. We know from a theorem of Hadamard that its natural domain of holomorphy is the open disc D = x C ; x 1 < 1 . The series of the derivatives of any order { ∈ | − | } (starting from order 0) converge uniformly on the closed disc D. The function f1 admits then an asymptotic expansion at 0 on any sector included in D.

Figure 2 3.1. PRESHEAVES AND SHEAVES 47

π π 1 Consider now the arc I =] 2 , 2 [ of S . To any θ I there is a sector θ = Iθ ]0,Rθ[ − ∈ × π on which f1 is well defined and belongs to ( θ). However, as θ approaches the A ± 2 radius Rθ tends to 0 and there is no sector = I ]0,R[ with R> 0 on which f1 is even × defined. Thus, Condition 2 of Definition 3.1.9 fails on U = I.

⊲ The sheaf over S1. — The sheaf of asymptotic functions over S1 is A A the sheaf associated with the presheaf . A section of over an interval I is A A defined by a collection of asymptotic functions f ( ) on = I ]0,R [ j ∈ A j j j × j where I is an open covering of I and R = 0 for all j. The sheaf is a { j} j 6 A sheaf of differential C-algebras. ⊲ The subsheaf <0 of flat germs. — Given an open sector = I ]0,R[ A I,R × (cf. Notations. 2.2.2), we define <0(I)= lim <0( ). −→ I,R A R→0 A The set <0(I) is a subset of (I). Considering the restriction maps ρ of A A J,I the presheaf (I) restricted to <0(I) we obtain a presheaf I <0(I) over A A 7→ A S1. The associated sheaf is denoted by <0 and is a subsheaf of over S1. A A ⊲ The Taylor map. — The Taylor map T : ( ) C[[x]] induces I,R A I,R → a map T : C[[x]] A → also called Taylor map which is a morphism of sheaves of C-differential algebras with kernel <0. Thus, <0 is a subsheaf of ideals of . A A A ⊲ The sheaf over S1. — Similarly, one defines a presheaf over S1 As As by setting (I)= lim ( ) s −→ s I,R A R→0 A for the set of (equivalence classes of) s-Gevrey asymptotic functions on a sector based on I. Its associated sheaf is denoted by . As ⊲ The sheaf ≤−k over S1. — One also defines a presheaf by setting A ≤−k(I)= lim ≤−k( ) −→ I,R A R→0 A and ≤−k denotes the associated sheaf over S1. According to Proposi- A tion 2.3.17, the presheaf ≤−k is a sub-presheaf of , and then, ≤−k is a A As A subsheaf of , precisely, the subsheaf of s-Gevrey flat germs. As The Taylor map T : C[[x]] induces a Taylor map A → T = T : C[[x]] s As −→ s 48 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY which is a morphism of sheaves of C-differential algebras with kernel ≤−k. A Thus, ≤−k is a subsheaf of ideals of . A As 3.1.6. Quotient sheaves and exact sequences. — From now on, unless otherwise specified, we suppose that all the sheaves or presheaves we consider are sheaves or presheaves of Abelian groups (or, more generally, sheaves or presheaves with values in an Abelian category ). Recall that such sheaves or C presheaves and their morphisms form themselves an Abelian category which will allow us to talk of exact sequences of sheaves.

Given a sheaf with values in and a subsheaf one deﬁnes a presheaf G C F by setting U (U)/ (U) for all open set U of the base space X, the 7→ G F restriction maps being induced by those of . G Condition 1 of sheaves is always satisﬁed (for a proof see [Mal95, An- nexe 1] for instance) while Condition 2 fails in general (cf. Exa. 3.1.24).

Definition 3.1.23.— One defines the quotient sheaf = / to be the H G F sheaf over X associated with the presheaf U (U)/ (U) for all open set U of X 7−→ G F with restriction maps induced by those of . G If and are sheaves of Abelian groups or of vector spaces so is the F G quotient . If is a sheaf of algebras and a subsheaf of ideals, then is a H G F H sheaf of algebras. As noticed at the end of Section 3.1.3, the fact that the quotient presheaf satisfies Condition 1 of sheaves (Def. 3.1.9) means that the natural map (U)/ (U) (U) G F →H is injective. If Condition 2 were also satisfied then this natural map would be surjective. However, this is not true, in general, as shown by the Exam- ple 3.1.24 below.

Example 3.1.24.— (Quotient sheaf and Euler equation) We saw in Example 2.2.4 that the Euler equation dy x2 + y = x (1) dx − +∞ e ξ/x admits an actual solution E(x)= 0 1+ξ dξ which is asymptotic to the Euler series E(x)= ( 1)nn!xn+1 on the sector 3π < arg(x) < + 3π . n≥0 − R − 2 2 e P 3.1. PRESHEAVES AND SHEAVES 49

Consider the homogeneous version of the Euler equation 2 3 d y 2 dy 0 y x +(x + x) y =0. E ≡ dx2 dx − Recall that one obtains the equation 0y = 0 by dividing equation(1) by x and differen- E tiating. In any direction, 0 y = 0 admits a two dimensional C-vector space of solutions E spanned by e1/x and E(x). 1 Following P. Deligne we denote by the sheaf over S of the germs of solutions of ( 0) V E having an asymptotic expansion at 0 and we denote by θ the stalk of in a direction θ. V V The sheaf is a sheaf of vector spaces and a subsheaf of seen as a sheaf of vector spaces. V A Since E(x) has an asymptotic expansion in all directions 3π/2 <θ< 3π/2 and e1/x has − an asymptotic expansion (equal to 0) on (x) < 0 we can assert that ℜ 2 if +π/2 <θ< 3π/2, dimC θ = V ( 1 if π/2 θ +π/2 − ≤ ≤ · Denote by <0 = <0 the subsheaf of flat germs of . We observe V V ∩A V that (S1)= 0 and <0(S1)= 0 , hence the quotient (S1)/ <0(S1)= 0 . V { } V { } V V { } A global section of the quotient sheaf / <0 is a collection of solutions over an V V open covering of S1 which agree on the intersections up to flat solutions. The solution E induces such a global section while e1/x does not. Thus, the space of global sections 1 <0 1 <0 Γ(S ; / ) has dimension dimC Γ(S ; / ) = 1. This shows that the quotient sheaf V V V V / <0 is different from the quotient presheaf. The quotient sheaf / <0 is isomorphic to V V V V the constant sheaf C as a sheaf of C-vector spaces.

Let f : be a morphism of sheaves with values in over the same F →G C base space X. Let ρ and ρ′ denote the restriction maps in and V,U V,U F G respectively. One can define the presheaves Ker (f),Im (f) and Coker (f) over X with values in by setting C ⊲ for Ker (f) : U ker f(U) : (U) (U) for all open set U X 7−→ F → G ⊆ with restriction maps r = ρ ; V,U V,U |ker(f(U)) ⊲ for Im (f) : U f (U) with restriction maps r′ = ρ′ ; 7−→ F V,U V,U |f(F(U)) ⊲ for Coker (f): U (U)/f (U) with restriction maps canonically 7−→ G F induced from ρ′ on the quotient. V,U So defined, Ker (f) and Im (f) appear as sub-presheaves of and respec- F G tively, Coker (f) as a quotient of . For a definition by a universal property G we refer to the classical literature. One can check that the presheaf Ker (f) is actually a sheaf (precisely, a subsheaf of ). Hence, the definition: F Definition 3.1.25.— The sheaves kernel, image and cokernel of a morphism of sheaves f can be defined as follows. 50 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

⊲ The kernel er(f) of the sheaf morphism f, is the sheaf deﬁned by K U ker f(U) for all open set U X 7−→ ⊆ with the restriction maps ρ . V,U |ker(f(U)) ⊲ The image m(f) and the cokernel oker(f ) of the sheaf morphism f, are I C the sheaves respectively associated with the presheaves Im (f) and Coker (f).

The sheaves oker(f ) and m(f) are respectively a quotient and a kernel: C I oker(f )= / m(f ), m(f )= er oker(f ) C G I I K G→C where oker(f ) stands for the canonical quotient map. G→C Definition 3.1.26.— Exactness of sequences of presheaves and of sheaves are defined by the following non-equivalent conditions: f g ⊲ A sequence of presheaves is said to be exact when F −→ G −→ H Im (f (U)) = Ker (g(U)) for all open set U X. f g ⊆ ⊲ A sequence of sheaves is said to be exact when F −→ G −→ H Im(f ) = Ker(g ) for all x X. x x ∈ fn fn+1 ⊲ A sequence n−1 n n+1 of presheaves or ··· → F −→ F −−→f F →f ··· sheaves is exact when each subsequence n n+1 is exact. Fn−1 −→ Fn −−→ Fn+1 A sequence of sheaves can be seen as a sequence of presheaves. One can show that exactness as a sequence of presheaves implies exactness as a sequence of sheaves the converse being false in general. Precisely, to a short (hence f g to any) exact sequence of presheaves 0 0 there corresponds → F → G f→ Hg → canonically the exact sequence of sheaves 0 0. Reciprocally, f g → F →G → H → an exact sequence 0 0 of sheaves can be seen as a sequence → F →G → H → f g of presheaves but, in general, only the truncated sequence 0 → F → G → H is exact as a sequence of presheaves. Let PreshX and ShX denote respectively the categories of presheaves and sheaves over X with values in a given Abelian category . In the language of C categories the properties above are formulated as follows. ⊲ The functor of sheafification Presh Sh is exact. X → X .The functor of inclusion Sh ֒ Presh is only left exact ⊳ X → X 3.1. PRESHEAVES AND SHEAVES 51

3.1.7. The Borel-Ritt Theorem revisited. — By construction, <0(I) A and ≤−k(I) are the kernels of the Taylor maps A T : (I) C[[x]] and T : (I) C[[x]] I A −→ s,I As −→ s respectively for any open arc I of S1. Hence, the sequences 0 <0 T C[[x]] and 0 ≤−k Ts C[[x]] → A −→ A −−→ → A −→ As −−→ s are exact sequences of presheaves and they generate the exact sequences of sheaves of differerential algebras 0 <0 T C[[x]] and 0 ≤−k Ts C[[x]] . →A −→ A −−→ →A −→ As −−→ s The Borel-Ritt Theorem 2.4.1 allows one to complete these sequences into short exact sequences as follows. Corollary 3.1.27 (Borel-Ritt).— The sequences (18) 0 <0 T C[[x]] 0, →A −→ A −−→ → (19) 0 ≤−k Ts C[[x]] 0 →A −→ As −−→ s → are exact sequences of sheaves of differential C-algebras over S1. Equivalently, the quotient sheaves / <0 and / ≤−k are isomorphic via the Taylor map A A As A to the constant sheaves C[[x]] and C[[x]]s respectively, as sheaves of differential C-algebras.

With this approach, the surjectivity of T or Ts means that, given any series and any direction there exist a sector containing the direction and a function asymptotic on it to the given series. We cannot not claim that the sector can be chosen to be arbitrarily wide. Observe that (18) and (19) are not exact sequences of presheaves over S1. Indeed, the range of the Taylor map T : (S1) C[[x]], as well as the range A → of T : (S1) C[[x]] , is made of convergent series and, consequently, these s As → s maps are not surjective.

3.1.8. Change of base space: direct image, restriction and extension by 0. — The following deﬁnition makes sense since for f continuous and U open in Y the set f −1(U) is open in X. Deﬁnition 3.1.28 (Direct image).— Let f : X Y be a continuous → map. With any sheaf over X one can associate a sheaf f over Y called F ∗F its direct image by setting f (U)= f −1(U) for all open set U in Y, ∗F F 52 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY with restriction maps

ρ (s )= ρ −1 −1 (s) for all open sets V U in Y. ∗ V,U ∗ f (V ),f (U) ⊆ When is a sheaf of Abelian groups, vector spaces, etc. . . , so is its direct F image f . ∗F To a morphism ϕ : of sheaves over X there corresponds a morphism F →G of sheaves ϕ : f f over Y defined by ∗ ∗F → ∗G s f (U)= f −1(U) ϕ(s ) f −1(U) = f (U). ∗ ∈ ∗F F 7−→ ∗ ∈G ∗G The functor direct image is left exact. Thus, to an exact sequence 0 ′ u ′ v ′′ → F −→ F −→ F there corresponds the exact sequence u v 0 f ′′ ∗ f ′ ∗ f ′′. → ∗F −→ ∗F −→ ∗F We suppose now that X is a subspace of Y with inclusion j : X ֒ Y and → that we are given a sheaf over Y . The restriction of into a sheaf over X is G G fully natural in terms of espaces étalés. We denote by π : G Y the espace → étaléassociated with . G Definition 3.1.29 (Restriction).— The sheaf restricted to X is the sheaf with espace étalé G G|X −1 π| : π (X) X. π−1(X) → The definition makes sense since as π : G Y is a local homeomorphism −1 → so is π| : π (X) X. The restricted sheaf can also be seen as the π−1(X) → inverse image of by the inclusion map j, a viewpoint which we won’t develop G here.

As in the previous section we consider now sheaves of Abelian groups and j we denote by 0 the neutral element. With X ֒ Y let and ′ be sheaves −→ F F over X and Y respectively.

Deﬁnition 3.1.30 (Extension).— ⊲ A sheaf ′ is an extension of a sheaf if its restriction ′ to X is F F F|X isomorphic to . F 3.1. PRESHEAVES AND SHEAVES 53

⊲ An extension ′ of is an extension by 0 if, for all y Y X, the ′ F F ′ ∈ \ stalk y is 0. (Equivalently, is the constant sheaf 0.) F F|Y \X Deﬁnition 3.1.31 (Support of a section).— The support of a section s Γ(U; ) is the subset of U where s does not ∈ F vanish: supp(s)= x U ; s = 0 . ∈ x 6 Example 3.1.32.— Let be the sheaf of C-vector spaces generated over S1 by the E function e1/x. The sheaf is isomorphic to the constant sheaf with stalk C over S1. Let E e(x) be the class of e1/x in the quotient sheaf / <0 where <0 = <0. Thus, e(x)=0 E E E E∩A for (x) < 0 and the support of e is the arc π/2 arg(x) π/2, a closed subset of S1. ℜ − ≤ ≤

The support supp(s) is always a closed subset of U, for, if a germ sx is 0 then, there is an open neighborhood Vx of x on which sx is represented by the 0 function generating thus the germs 0 on a neighborhood of x. Recall that a subset X of Y is said to be locally closed in Y if any point x X admits in Y a neighborhood V (x) such that its intersection V (x) X ∈ Y Y ∩ is closed in VY (x). This is equivalent to saying that there exist X1 open in Y and X closed in Y such that X = X X . 2 1 ∩ 2 Deﬁnition 3.1.33 (Sheaf j!F).— Suppose X is locally closed in Y and denote by j : X ֒ Y the inclusion map → of X in Y . Given a sheaf of Abelian groups over X one deﬁnes the sheaf j F !F over Y by setting, for all open U of Y , j (U)= s Γ(X U; ); supp(s) is closed in U !F ∈ ∩ F with restriction maps induced by those of j (of which j is a subsheaf). ∗F !F One can check that j is a sheaf; it is then clearly a subsheaf of j and !F ∗F there is a canonical inclusion j ֒ j . Moreover, j is an extension of !F → ∗F !F by 0. When X is closed in Y then the two sheaves coincide: j = j . F !F ∗F Unlike the functor j∗ which is only left exact, the functor j! is exact. The extension of sheaves by 0 provides a characterization of locally closed subspaces as follows: X is locally closed in Y if and only if, for any sheaf F over X, there is a unique extension of to Y by 0 (cf. [Ten75] Thm. 3.8.6). F Example 3.1.34 (j = j ).— As an illustration consider the sheaf ′ generated as ∗ ! E a sheaf of C-vector spaces6 by e1/x over the punctured disc D∗ = x C ;0 < x < 1 { ∈ | | } .and consider the inclusion j : D∗ ֒ C → 54 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

′ ′ The direct image j∗ of by j is a non-constant sheaf of C-vector spaces. Indeed, E E ′ n for U a connected open set in C, one has j∗ (U) C where n is the number of connected E ≃ components of U D∗. ∩

Figure 3

′ The stalks of j∗ are given by E ∗ ′ C if x D , j∗ x ∈ E ≃ ( 0 otherwise, ′ ′ ∗ so that, in some way, the direct image j∗ spreads out, onto the closure of D . Thus, ′ E′ E the direct image j∗ is an extension of but not an extension by 0. E ′ E ′ ∗ On the contrary, the sheaf j! is an extension of by 0. It is well deﬁned since D E E ′ ′ being open in C is also locally closed in C. This shows that j∗ = j! and therefore, that E 6 E the functors j∗ and j! are diﬀerent.

3.2. Cechˇ cohomology Let be a sheaf over a topological space X. We assume that is a F F sheaf of Abelian groups. The set Γ(U; ) of sections of over a U X is F F ⊂ then naturally endowed with a structure of Abelian group and ( ) = 0 , F ∅ { } the trivial Abelian group 0. Unless otherwise speciﬁed, all the coverings we consider are coverings by open sets.

3.2.1. Cechˇ cohomology of a covering .— Let = U be an U U { i}i∈I open covering of X. Denote U = U U , U = U U U , and so on. . . i,j i ∩ j i,j,k i ∩ j ∩ k Definition 3.2.1.— One defines the Cechˇ complex of associated with the F covering to be the differential complex U d0 d1 0 Γ(Ui ; ) Γ(Ui ,i ; ) → i0 0 F −→ i0,i1 0 1 F −→ d d d Q n−1 Γ(UQ ; ) n Γ(U ; ) n+1 ··· −−→ i0,...,in F −→ i0,...,in+1 F −−−→ · · · where, for all n, the mapQdn is defined by Q d : f =(f ) g =(g ) n i0,...,in 7−→ i0,...,in+1 3.2. CECHˇ COHOMOLOGY 55 where n+1 ℓ gi0,...,in+1 = ( 1) f − i0,...,iℓ−1,ıℓ,iℓ+1,...,in+1 b Ui ,...,i Xℓ=0 0 n+1 the hat over i indicating that the index i is omitted. ℓ ℓ Each term of the complex is an Abelian group.

The maps dn are morphisms of Abelian groups. Consequently, the image im dn and the kernel ker dn are Abelian groups. For all n, the maps dn are “diﬀerentials” which, in this context, means that d d = 0 and thus, n ◦ n−1 im d ker d and the quotients ker d /im d are Abelian groups. n−1 ⊂ n n n−1 Deﬁnition 3.2.2.— One calls ⊲ n-cochains of (with values) in the elements of the Abelian group U F n( ; )= Γ(U ; ), C U F i0,...,in F ⊲ n-cocycles of (with values)Y in the elements of the Abelian group U F n( ; ) = ker d , Z U F n ⊲ n-coboundaries of (with values) in the elements of the Abelian U F group n( ; )=im d , B U F n−1 ⊲ n-th Cechˇ cohomology group of (with values) in the Abelian group U F Hn( ; )= n( ; )/ n( ; ) = ker d /im d . U F Z U F B U F n n−1 In particular, H0( ; ) Γ(X; ) the set of global sections of over X. U F ≃ F F

Definition 3.2.3 (Refinement of a covering).— A covering = V V is said to be finer than the covering = U , and we de- { j}j∈J U { i}i∈I note , if any element in is contained in at least one element of . VU V U Equivalently, one can say that there exists a map σ : J I such that V U for all j J. −→ j ⊂ σ(j) ∈ Such a map is called inclusion map or simplicial map. With the simplicial map σ are naturally associated the maps σ∗ : n( ; ) n( ; ), f =(f ) σ∗f =(F ) n C U F −→ C V F i0,...,in 7−→ n j0,...,jn given by Fj ,...,j = f V . 0 n σ(j0),...,σ(jn)| j0,...,jn 56 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

∗ ∗ ˇ The family σ =(σn) deﬁnes a morphism of Cech complexes and induces, for all n, a morphism of groups Sn( , ): Hn( ; ) Hn( ; ). V U U F −→ V F It turns out that these latter homomorphims are independent of the choice of the simplicial map σ. The case when n = 1 has the following speciﬁcity:

Proposition 3.2.4.— When n = 1, the morphism S1( , ) : H1( ; ) H1( ; ) is injective. V U U F −→ V F We refer to [Ten75, Thm. 4.15, p. 148] .

3.2.2. Cechˇ cohomology of the space X.— The preceding section suggests to take the direct limit (cf. Def. p. 40) of the groups Hn( ; ) U F using the maps S( , ) as the coverings become finer and finer. Indeed, the V U coverings of X endowed with fineness form an ordered, right filtrant(1) “set” and the maps Sn( , ) satisfy, for all n, the conditions: V U

⊲ Sn( , ) = Id for all , U U U ⊲ Sn( , ) Sn( , )= Sn( , ) for all W V ◦ V U W U WVU providing thus a direct system (Hn( ; ), Sn( , )) of Abelian groups. U F V U The only problem is that coverings of a topological space do not form a set. That difficulty can be circumvented by limiting the considered coverings to those that are indexed by a given convenient set, i.e., a set of indices large enough to allow arbitrarily fine coverings. In the cases we consider any count- able set is convenient, say, N or Z. Actually, for X = S1, we may consider coverings with a finite number of open sets since there exists finite coverings of S1 that are arbitrarily fine. From now on, we assume that the coverings are indexed by subsets J of N. Another trick due to R. Godement consists in considering only the coverings U indexed by the points x X with the condition x U (cf. [God58, { x} ∈ ∈ x Sect. 5.8, p. 223]). Hence, the following definition:

Deﬁnition 3.2.5.— The n-th Cechˇ cohomology group of the space X (with values) in is the direct limit of the cohomology groups Hn( ; ), the limit F U F

(1) “Right filtrant” means here that to each finite family U1,..., Up of open coverings of X there is a covering V finer than all of them. 3.2. CECHˇ COHOMOLOGY 57 being taken over coverings ordered with fineness. One denotes Hn(X; ) = lim Hn( ; ). −→ F U U F When X is a manifold and n> dim X there exists arbitrarily fine coverings without intersections n +1 by n + 1 and then, Hn(X; ) = 0. The canonical F isomorphism H0(X; ) Γ(X; ) is valid without restriction. F ≃ F The following two results are useful. Theorem 3.2.6 (Leray’s Theorem).— Given an acyclic covering of X U which is either closed and locally finite or open then, Hn( ; )= Hn(X; ) for all n. U F F Acyclic means that Hn(U ; ) = 0 for all U and all n 1. i F i ∈U ≥ We refer to [God58, Thm. 5.2.4, Cor. p. 209], (case closed and locally U finite) and to [God58, Thm. 5.4.1, Cor. p. 213] (case open). U Theorem 3.2.7.— To any short exact sequence of sheaves of Abelian groups over X 0 0 → G −→ F −→ H → there is a long exact sequence of cohomology 0 H0(X; ) H0(X; ) H0(X; ) → G −→ F −→ H δ 0 H1(X; ) H1(X; ) H1(X; ) −→ G −→ F −→ H δ 1 H2(X; ) H2(X; ) H2(X; ) −→ G −→ F −→ H δ2 −→···

The maps δ0,δ1,... are called coboundary maps. For their general deﬁni- tion, see the references above.

3.2.3. The Borel-Ritt Theorem and cohomology. — We know from Corollary 3.1.27 that the sheaves / <0 and / ≤−k are constant sheaves A A As A with stalks C[[x]] and C[[x]]s respectively. Their global sections Γ(S1; / <0) H0(S1; / <0) and Γ(S1; / ≤−k) H0(S1; / ≤−k) A A ≡ A A As A ≡ As A are then also respectively isomorphic to C[[x]] and C[[x]]s and we can state the following corollary of the Borel-Ritt Theorem. 58 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

Corollary 3.2.8 (Borel-Ritt).— The Taylor map induces the following isomorphisms:

H0(S1; / <0) C[[x]], H0(S1; / ≤−k) C[[x]] . A A ≃ As A ≃ s We can synthesize:

(equivalence class of a) formal series 1  0-cochain (fj)j∈J over S f(x)= a xn C[[x]]   with values in and n ∈ ⇐⇒  A n≥0   coboundary (fj fℓ)j,ℓ∈J X −<0 e with values in   A   The components fj(x) of the 0-cochains are all asymptotic to f(x).

(equivalence class of a) e s-Gevrey series 1 0-cochain (fj)j∈J over S f(x)= a xn C[[x]]  with values in s and n ∈ s ⇐⇒  A n≥0  coboundary (fj fℓ)j,ℓ∈J X −≤−k e with values in   A   The components fj(x) of the 0-cochains are all s-Gevrey asymptotic to f(x). Due to Proposition 2.3.17 it would actually be suﬃcient to ask for the coboundary to be with values in <0. This latter equivalence will be improved A ine Corollary 6.2.2.

3.2.4. The case when X = S1 and the Cauchy-Heine Theorem. — Since, in what follows, we will mostly be dealing with sheaves over S1, it is worth developing this case further. With X = S1 things are often made simpler by the fact that S1 is a manifold of dimension 1. On another hand, 1 one has to take into account the fact that S has a non-trivial π1.

Deﬁnition 3.2.9 (Good covering).— An open covering = (I ) of I j j∈J S1 is said to be a good covering if ⊲ it is ﬁnite with J = p elements, | | 1 ⊲ its elements Ij are connected (i.e., open arcs of S ), ⊲ it has thickness 2 (i.e., no 3-by-3 intersections), ≤ 3.2. CECHˇ COHOMOLOGY 59

1 ⊲ when p = 2 its two open arcs I1 and I2 are proper arcs of S so • that I1 I2 is made of two disjoint open arcs which we denote by I1 and • ∩ I2; when p 3 its open arcs Ij can be indexed by the cyclic group Z/pZ so • ≥ that I := I I = and I I = as soon as ℓ j > 1 modulo p. j j ∩ j+1 6 ∅ j ∩ ℓ ∅ | − | The deﬁnition implies that open arcs of a good covering are not nested. • The family of the arcs I is sometimes called the nerve of the covering . j I

The case p = 1, i.e., the case of coverings of S1 by just one arc, is worth to consider. These unique arcs cannot be proper arcs of S1: one has to introduce overlapping arcs i.e., arcs of the universal cover of S1 of length > 2π. Such coverings are widely used to make proofs simpler by using the additivity of 1-cocycles. A typical example is given by the Cauchy-Heine Theorem (Thm. 2.5.2 and Cor. 3.2.14 below).

Deﬁnition 3.2.10 (Elementary good covering).— An open covering = I with only one overlapping open arc I = ]α,β + 2π[ and nerve I• { } I =]α,β[( S1 is called an elementary good covering. Example 3.2.11 (The Euler series as a 0-cochain)

The Euler series f(x), which belongs to C[[x]]1, can be seen as a 0-cochain as follows. 1 Consider the covering = I1,I2 of S made of the arcs e I { } I1 =] 3π/2, +π/2[ and I2 =] π/2, +3π/2[. − − The elements of intersect over the two arcs •I • 1 1 I 1 = x S ; (x) < 0 and I 2 = x S ; (x) > 0 . { ∈ ℜ } { ∈ ℜ } The corresponding 0-cochain to consider is the pair (f1(x),f2(x)) made of the restrictions

of the Euler function f(x) to I1 and I2 respectively. Both f1(x) and f2(x) are sections of • • 1. The coboundary (f 1, f 2) is given by A • • • • 1/x f (x)= f1(x) f2(x)=2πie on I 1 and f (x)= f2(x) f1(x)=0 on I 2 1 − 2 − and has values in ≤−1. A

Figure 4 60 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

• Since the component f is trivial one could also consider a branch covering made of 2 • the unique arc I =] 3π/2, +3π/2[ overlapping on I = x S1 ; (x) < 0 . − { ∈ ℜ }

Figure 5

The (branched) 0-cochain f(x) is 1-Gevrey asymptotic to f(x) on I and its cobound- • ary f +(x) f −(x)=2πi exp(1/x) is a section of ≤−1 over I . − A e Given a good covering = I of S1, a 1-cochain is a family I { j} f Γ(I I ; ) for j and ℓ Z/pZ. j,ℓ ∈ j ∩ ℓ F ∈ • • • The 1-cocycle conditions f +f = f on I I I for all j,k,ℓ are empty j,k k,ℓ j,ℓ j ∩ k ∩ ℓ since so are the 3-by-3 intersections; consequently, any 1-cochain is a 1-cocycle. Taking into account the necessary conditions fj,j = 0 and fk,j = fj,k on 1- • −• cocycles, a 1-cocycle can thus be seen as any collection (f Γ(I ; )) for j ∈ j F j Z/pZ. ∈ • By linearity, a 1-cocycle (f j)j∈Z/pZ can be decomposed into a sum • • j∈Z/pZ ϕj where ϕj is the 1-cocycle over the covering having all trivial I • th Pcomponents (equal to 0, the neutral element) but the j equal to f j. Fix • j and consider the elementary good covering j whose nerve is Ij and the • • I 1-cocycle f j Γ(Ij, ). The covering is finer than j. We identify the • ∈ • F I • I 1-cocycles ϕj and f j and we say that the 1-cocycle ϕj can be lifted into the • elementary 1-cocycle f j. Proposition 3.2.12.— There exist arbitrarily fine good coverings of S1 Consequently, when is a sheaf over S1, to determine H1(S1; ) it suffices F F to consider good coverings.

Example 3.2.13.— (Euler equation and cohomology) We consider the elementary good covering = I of S1 deﬁned by the overlapping interval I = ] I { }• − 3π/2, +3π/2[ with self-intersection I = ] 3π/2, π/2[ and we consider the sheaf of − − V asymptotic solutions of the Euler equation (cf. Exa. 3.1.24). A 1-cocycle of in is • • I V a section ϕ (x)= af(x)+ be−1/x over I with arbitrary constants a and b in C. There is no 1-cocycle condition. The 0-cochains are of the form cf(x) over I, with c C an ∈ arbitrary constant and they generate the 1-coboundaries c f(xe2πi) f(x) =2πice−1/x. − 3.2. CECHˇ COHOMOLOGY 61

• The cohomological class of ϕ is then uniquely represented by af(x) for 3π/2 < arg(x) < − π/2. Hence, H1( ; ) is a vector space of dimension one, isomorphic to C. − I V Given a covering of S1 ﬁner than we have seen (cf. Prop. 3.2.4) that the J 1 1 I map SJ ,I : H ( ; ) H ( ; ) is injective. Let us check that it is surjective on the I V → J V example of = J1,J2 for J1 =] π/4, 5π/4[ and J2 =] 5π/4,π/4[. • J { } • − − We set J 1= ]3π/4, 5π/4[ and J 2=] π/4,π/4[. • • − • • A 1-cocycle (ϕ1, ϕ2) over the covering is cohomologous to (ϕ1 + ϕ2, 0) via the • J • • • 0-cochain (0, ϕ2) where we keep denoting by ϕ2 the continuation of ϕ2 to J 2. How- • • • • • ever, ϕ=ϕ1 + ϕ2 can be continued to I (we keep denoting by ϕ the continuation) and • • • therefore, the 1-cocycle (ϕ1 + ϕ2, 0) lifts up into the 1-cocycle ϕ of the covering . I

Figure 6

The proof extends to any good covering ﬁner than by induction on the number J I of connected 2-by-2 intersections. We can conclude that H1( ; )= H1(S1; ). I V V The same result can be seen as a consequence of the theorem of Leray (Thm. 3.2.6) after showing that is acyclic for . I V In the case when X = S1 the long exact sequence of cohomology of Theo- rem 3.2.7 reduces to 0 H0(S1; ) H0(S1; ) H0(S1; ) → G −→ F −→ H δ 0 H1(S1; ) H1(S1; ) H1(S1; ) 0. −→ G −→ F −→ H → The coboundary map δ is deﬁned as follows: The sheaf is the quotient 0 H of by . A 0-cocycle in H0(S1; ) is a collection of f Γ(I ; ) such F G H i ∈ i F that f f belong to Γ(I ; ) for all i,j. There corresponds the 1-cocycle i − j i,j G (g = f f ) of with values in . To different representatives f ′ of i,j i − j i,j I G i the 0-cocycle there correspond a cohomologous 1-cocycle (g′ = f f ) of i,j i − j i,j with values in ; hence, an element of H1 I ; and consequently, an I G { i} G element of H1(S1; ). G The Cauchy-Heine Theorem (Thm. 2.5.2) can be reformulated as a cohomological condition as follows. Corollary 3.2.14 (Cauchy-Heine).— 62 CHAPTER 3. SHEAVES AND CECHˇ COHOMOLOGY

(i) The natural map H1(S1, <0) H1(S1, ) is the null map. A → A (ii) The natural map H1(S1, ≤−k) H1(S1, ) is the null map. A → As Proof. — (i) It suﬃces to prove the assertion for any good covering. Given a covering of S1 there is a natural map from H1( ; <0) into H1( ; ) I I A I A (cohomologous 1-cocycles of H1( ; <0) are also cohomologous in H1( ; )). I A I A By linearity, it suﬃces to consider the case of an elementary good covering • = I with self-intersection I (cf. Def. 3.2.10). The Cauchy-Heine Theorem I { } as stated in Thm. 2.5.2 says that a 1-cocycle of H1( ; <0) is a coboundary I A in H1( ; ), that is, it is cohomologous to the trivial 1-cocycle 0 in H1( ; ). I A I A (ii) Same proof by replacing <0 by ≤−k and by . A A A As Although the maps are zero maps, far from being null spaces, H1(S1, ) A and H1(S1, ) are huge spaces. As CHAPTER 4

LINEAR ORDINARY DIFFERENTIAL EQUATIONS: BASIC FACTS AND INFINITESIMAL NEIGHBORHOODS OF IRREGULAR SINGULARITIES

In this chapter, we first gather some basic facts on linear ordinary differential equations. Our aim is not to be exhaustive (in particular, we omit most of the proofs) but to provide the useful material to better understand series solutions of differential equations and examples. We end the chapter with the construction of infinitesimal neighborhoods for the singularities of solutions of linear differential equations at an irregular singular point in the spirit of the infinitesimal neighborhoods of algebraic geometry. The adequacy of such neighborhoods to characterize the summability properties of the formal solutions of a given differential equation is presented in Chapters 6 and 8 (Defs. 6.4.1 and 8.7.1). Consider a linear differential operator of order n dn dn−1 (20) D = b (x) + b (x) + + b (x) where b (x) 0 n dxn n−1 dxn−1 ··· 0 n 6≡ with analytic coefficients at x = 0. Unless otherwise specified, we assume that the coefficients bn,bn−1,...,b0 do not vanish simultaneously at x = 0. When the coefficients bn,bn−1,...,b0 are polynomials in x their maximal degree is called the degree of D.

4.1. Equation versus system With the differential equation Dy = 0, setting y y′ Y =  .  , .    y(n−1)      64 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS is associated its companion system ∆Y = 0 defined by the n-dimensional order one differential operator 0 1 0 ··· ...... d  . . . .  ∆= B(x) where B(x)= . dx − . 01 ·    b0 bn−1     − bn ··· ··· − bn  Reciprocally, the question is to determine if and how one can put a given system in companion form.

Deﬁnition 4.1.1 (Gauge transformation).— Given a system of dimension n with meromorphic coeﬃcients ∆Y dY B(x)Y = 0 a gauge transfor- ≡ dx − mation is a linear change of the unknown variables Z = TY with T invertible in a sense to be made precise. In the case when T belongs to GL(n, C x [1/x]) { } the gauge transformation T is said to be meromorphic; in the case when T belongs to GL(n, C[[x]][1/x]) it is said to be formal (meromorphic).

A gauge transformation Z = TY changes the system ∆Y = 0 into the diﬀerential system T∆Z = 0 with d dT T∆= T ∆T −1 = T −1 TBT −1. dx − dx − When T is meromorphic (resp. formal), so is T∆; however, T∆ may be meromorphic for some formal T . We can now answer the question.

Proposition 4.1.2 ((Deligne’s) Cyclic vector lemma) To any system ∆Y = 0 with meromorphic coeﬃcients there is a meromorphic gauge transformation Z = TY such that the transformed system T∆ = 0 is in companion form.

The formulation in terms of cyclic vectors (cf. Rem. 4.2.6) is due to P. Deligne [Del70, Lem II.1.3] although more algorithmic proofs already existed [Cop36], [Jac37]. The companion form is obtained by diﬀerential elimination. Despite the fact that the program is short and simple it is not (at least, not yet) available in computer algebra systems such as Mathematica or Maple (see [BCLR03] for a sketched algorithm and references; see also [Ram84, Thm. 1.6.16]). As a consequence of the Cyclic vector lemma, theoretical properties can be proved equally on equations or systems (as long as these properties stay unchanged under meromorphic gauge transformations). To perform calculations one could, in principle using the algorithm, go from equations to systems 4.2. THE VIEWPOINT OF D-MODULES 65 and reciprocally at convenience. Actually, these algorithms are usually very “expensive” and used sparingly.

4.2. The viewpoint of D-modules The notion of differential module, or equivalently, of -module generalizes D the notion of order one differential system in an abstract setting free of coordinates. From this viewpoint, the gauge transformations and the meromorphic or formal equivalence arise naturally. Suppose we are given a differential field (K,∂). Precisely, for our purpose, we suppose that K is either the field C x [1/x] of meromorphic series at 0 { } or the field C[[x]][1/x] of the formal ones. The derivation is ∂ = d/dx. The constant subfield C of K, i.e., the set of the elements f K satisfying ∂f = 0, ∈ is C = C and the C-vector space of the derivations of K has dimension 1 and generator ∂.

4.2.1. -modules and order one differential systems. — D Definition 4.2.1.— A differential module(1) (M, ) of rank n over K is a ∇ K-vector space M of dimension n equipped with a map : M M, ∇ −→ called connection, which satisfies the two conditions: (i) is additive; ∇ (ii) satisfies the Leibniz rule (fm) = ∂f m + f (m) for all f K ∇ ∇ · ∇ ∈ and m M. ∈ We may observe that is also C-linear. Indeed, when f C is a constant ∇ ∈ the Leibniz rule reads (fm)= f (m). ∇ ∇ The link with differential systems is as follows. Choose a K-basis e =[e e e ] of M and let 1 2 ··· n [ε ε ε ]= [e e e ]B with B gl(n,K) 1 2 ··· n − 1 2 ··· n ∈ be its image by (the minus sign is introduced to fit the usual notations for ∇ systems and has no special meaning). The connection is fully determined n ∇ by the matrix B. Indeed, let y = j=1 yjej be any element of M. In matrix notation, we write y = eY where Y = t y y is the column matrix of the P 1 ··· n (1) In French, one says “un vectoriel àconnexion”. 66 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS components of y in the basis e. Then, applying the Leibniz rule, we see that y is uniquely determined by ∇ y = e (∂Y BY ). ∇ − Thus, with the connection and the K-basis e is naturally associated the ∇ differential operator ∆ = ∂ B of order one and dimension n. − Definition 4.2.2.— Let (M , ) and (M , ) be differential modules. 1 ∇1 2 ∇2 (i) A morphism of differential modules from (M , ) to (M , ) is a 1 ∇1 2 ∇2 K-linear map : M M which commutes to the connections and , T 1 → 2 ∇1 ∇2 i.e., such that the following diagram commutes:

M T M 1 −−−−→ 2 ∇1 ∇2  T  M1 M2 y −−−−→ y (ii) A morphism is an isomorphism if is bijective. T T Denote by n and n the rank of (M , ) and (M , ) respectively. 1 2 1 ∇1 2 ∇2 Choose K-basis e1 and e2 of M1 and M2 and denote by ∆1 and ∆2 the differential system operators associated with of and in the basis e and e ∇1 ∇2 1 2 respectively. Denote by T the matrix of in these basis. The definition says T that is a morphism if T satisfies the relation T

∆2T = T ∆1.

It says that is an isomorphism if, in addition, n = n and the matrix T is T 1 2 invertible so that the condition may be written

−1 ∆1 = T ∆2T

−1 −1 −1 and is also valid for T in the form ∆1T = T ∆2; hence, the commutation of the diagram with T : M M replaced by T −1 : M M . We recognize 1 → 2 2 → 1 the formula linking the operators ∆1 and ∆2 under the gauge transformation (cf. Def. refgauge). Suppose M = M =: M. An invertible K-morphism T 1 2 T is just a change of K-basis in M. Therefore, to the connection there are the ∇ inﬁnitely many system operators T −1∆T associated with all T GL(n,K) ∈ and it is natural to set the following deﬁnition. 4.2. THE VIEWPOINT OF D-MODULES 67

Definition 4.2.3.— Two differential system operators ∆ = ∂ B 1 − 1 and ∆ = ∂ B are said to be K-equivalent if there exists a gauge transfor- 2 − 2 mation T in GL(n,K) such that −1 ∆1 = T ∆2T. When K = C x [1/x] is the field of meromorphic series the systems are said { } to be meromorphically equivalent. When K = C[[x]][1/x] is the field of formal meromorphic series they are said to be formally equivalent or formally meromorphically equivalent.

In modern language, we should say K-similar but the old denomination K- equivalent is still in common use. The condition is clearly an equivalence relation: indeed, any system op- −1 −1 −1 erator ∆ satisfies ∆ = I ∆I; if ∆1 = T ∆2T then ∆2 = S ∆1S with −1 −1 −1 −1 S = T ; if ∆1 = T ∆2T and ∆2 = S ∆3S then ∆1 = (ST ) ∆3(ST ). With this definition, a differential module can be identified to an equivalence class of systems. Denote by = K[∂] the ring of differential operators on K, i.e., the ring D of polynomials in ∂ with coefficients in K satisfying the non-commutative rule ∂x = x∂ + 1. Let us now show how a differential module can be identified to a -module, D i.e., a module over the ring in the classical sense. For this, we go to a dual D approach as follows. Consider n as a left -module and denote by ε =[ε ε ] its canonical D D 1 ··· n -basis. Given a n-dimensional system operator ∆ = ∂ B with coefficients D − in K we make it act linearly on n to the right by setting D n P ε [P P ]∆=[P ∂ P ∂] [P P ]B. j j 7−→ 1 ··· n 1 ··· n − 1 ··· n Xj=1 The cokernel n/ n∆ has a natural structure of left -module (but no natural D D D structure of right-module over ) and has rank n (its dimension as K-vector D space). Denote by M n/ n∆ this K-vector space of dimension n. The ≡D D images in the cokernel of the n elements ε1,...,εn — which we keep denoting by ε ,...,ε — of the canonical -basis ε form a K-basis of M. On another 1 n D hand, the operator ∂ acting on M to the left defines a connection on M: indeed, it acts additively and satisfies the Liebniz rule. The question remains to determine which class of systems it represents. From the relation ∂ B = 0 − 68 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

in M we deduce that, for all j = 1,...,n, the components of ∂εj in the basis ε are given by the jth row of the matrix B. Hence,

∂[ε ε ε ]=[ε ε ε ] tB. 1 2 ··· n 1 2 ··· n And we can conclude that the system operator associated with the connection ∂ is the adjoint ∆∗ = ∂ + tB of ∆. We can state:

Proposition 4.2.4.— Given a differential system operator ∆= ∂ B with − coefficients B in K the pair (M = n/ n∆,∂) defines a differential module D D of rank n over K with connection ∂ = ∗ adjoint to ∆. ∇ From now on, we may talk of the differential module n/ n∆, the connec- D D tion = ∂ being understood. With this result we can identify left -modules ∇ D and differential modules equipped with a K-basis. Observe, in particular, that a morphism or an isomorphism

φ : n/ n∆ n/ n∆ D D 1 −→ D D 2 in the sense of Deﬁnition 4.2.2 is a morphism or an isomorphism of -modules D in the classical sense and reciprocally.

Proposition 4.2.5.— Two system operators ∆ = ∂ B and ∆ = ∂ B 1 − 1 2 − 2 with coeﬃcients in K are K-equivalent if and only if the -modules n/ n∆ D D D 1 and n/ n∆ are isomorphic. D D 2

Proof. — We have to prove that two diﬀerential systems ∆1Y = 0 and ∆2Y = ∗ ∗ 0 on one hand and their adjoints ∆1Y = 0 and ∆2Y = 0 on the other hand are simultaneously K-equivalent. To this end, consider fundamental solutions and of ∆ Y = 0 and ∆ Y = 0 respectively in any convenient extension Y1 Y2 1 2 of K (for instance, the formal fundamental solutions given by Thm. 4.3.1). The systems ∆1Y = 0 and ∆2Y = 0 are equivalent if and only if there exists a −1 gauge transformation T GL(n,K) such that ∆1 = T ∆2T or equivalently ∈ t −1 t −1 t −1 2 = T 1. This latter relation is equivalent to the relation 2 = T 1 . Y Y t −1 t −1 Y Y Hence the result since 1 and 2 are fundamental solutions of the adjoints ∗ Y ∗ Y equations ∆1Y = 0 and ∆2Y = 0 respectively.

Remark 4.2.6. — Let us end this section with a remark on the Cyclic vector lemma (Prop. 4.1.2). In a diﬀerential module (M, ) of rank n one calls cyclic ∇ vector any vector e M such that the n vectors e, e,..., n−1e form a ∈ ∇ ∇ K-basis of M. In such a basis, the matrix of the connection reads in the ∇ 4.2. THE VIEWPOINT OF D-MODULES 69 form

0 0 an−1 ··· . .  1 . .  B∇ = ......  . . .   0 1 a0   ···    Let ∆ be a system of dimension n with coeﬃcients in K and consider the -module n/ n∆. In a cyclic basis e the system ∆ admits tB as matrix D D D − ∇ which is a companion form (cf. Sect. 4.1) but, stricto sensu, the minus signs in the sup-diagonal of 1’s. One can cancel these minus signs by taking the basis (e, ∂e,..., ( 1)n−1∂n−1e). − −

4.2.2. -modules and differential operators of order n.— The aim D of this section is to describe the K-equivalence of order n linear differential operators with coefficients in K. Consider a single linear differential operator

D = ∂n + b (x)∂n−1 + + b (x), b ,...,b K. n−1 ··· 0 0 n−1 ∈ The operator D acts linearly on by multiplication to the right. Its cokernel D / D has a natural structure of left -module. The pair ( / D,∂) defines D D D D D a differential module of rank n. Again, by abuse, we talk of the differential module / D, the connection ∂ being understood. D D Proposition 4.2.7.— Let ∆ be the companion system operator of D (cf. Sect. 4.1). Then, the -modules / D and n/ n∆ are isomorphic. D D D D D Proof. — Consider the map

U : n , (δ δ ) δ + δ ∂ + + δ ∂n−1 D −→ D 1 ··· n 7−→ 1 2 ··· n and the map V : n , projection over the last component, deﬁned by D →D (δ δ ) δ . 1 ··· n 7−→ n The maps U and V are -linear; the diagram D n ·∆ n D −−−−→D V U  ·D    Dy −−−−→ Dy 70 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS commutes and it can be completed into the commutative diagram with exact rows 0 n ·∆ n n/ n∆ 0 −−−→D −−−−→D −−−−→D D −−−→ V U u  ·D   0   / D 0. −−−→ Dy −−−−→ Dy −−−−→ D yD −−−→ The quotient map u does exist. It is left -linear and surjective since U is also D left -linear and surjective. On the other hand, the modules n/ n∆ and D D D / D have equal ranks. Therefore, u is an isomorphism of K-vector spaces D D and in particular, is injective. From Propositions 4.2.7 and 4.2.5 we may set the following deﬁnition.

Definition 4.2.8 (equivalent operators).— Two linear differential operators D and D of order n are said to be K-equivalent if the -modules 1 2 D / D and / D are isomorphic. D D 1 D D 2 Let us now make explicit the equivalence of order n linear differential operators in the spirit of Definition 4.2.3. Recall that = K[∂] is a non commutative ring with non-commutation D relations generated by ∂x = x∂ +1. In the ring there is an euclidian division D on the right and on the left. Consequently, any left or right ideal is principal and any two differential operators have a greatest common divisor on the left (denoted by lgcd) and on the right (rgcd) as well as a least common multiple on the left (llcm) and on the right (rlcm). These gcd and lcm are uniquely determined by adding the condition that they are monic polynomials, which we do. The counterpart for a differential operator D of a gauge transforma- ∈D tion for a system involves a transformation , with A , of the form TA ∈D (D) = llcm(D,A)A−1. TA By this, we mean that we take the lcm of D and A on the left and we divide it by A on the right (this is possible since, by definition, A can be factored on the right in any llcm involving A). In other words, (D) is the factor of TA smallest degree we must multiply A on the left to obtain a left multiple of D. Notice that such a factor is unique due to the uniqueness of llcm(D,A) as a monic polynomial.

Proposition 4.2.9.— The diﬀerential operators D and D are K- 1 2 ∈ D equivalent if and only if there exists A prime to D to the right such ∈ D 2 4.3. CLASSIFICATIONS 71 that D = (D ). 1 TA 2 We may notice that, as A and D are prime, the operators D and (D ) 2 2 TA 2 have the same order.

Proof. — By definition, the K-equivalence of D1 and D2 means that there is an isomorphism of -modules D ϕ : / D / D . D D 1 −→ D D 2 As a morphism of -modules the map ϕ is well defined by D ϕ(1 + D )= A + D . D 1 D 2 For any L , one has then ϕ(L + D ) = LA + D . Since ϕ(D ) = 0 ∈ D D 1 D 2 1 there exists L such that D A = L D . Conversely, any A such that 1 ∈ D 1 1 2 there is an L satisfying D A = L D determines a morphism of -modules 1 1 1 2 D from / D into / D by setting ϕ(1 + D )= A + D . D D 1 D D 2 D 1 D 2 The injectivity of ϕ means that the condition ϕ(L) = 0, i.e., LA = PD2 for a certain P , implies L = QD with Q . Hence, to any rela- ∈D 1 ∈ D tion LA = PD there is Q such that PD = QD A, that is to say, any 2 ∈ D 2 1 left common multiple of A and D2 is a left multiple of D1A. Otherwise said, D1A is the llcm of A and D2 and then, D = (D ). 1 TA 2 Let us now express the surjectivity of ϕ. This amount to the fact that there exists L such that ϕ(L + D )=1+ D , which means that there ∈D D 1 D 2 is P such that LA + PD = 1. This is a Bézout relation for A and D on ∈D 2 2 the right which means that A and D2 are prime on the right.

4.3. Classifications We denote by K = C[[x]][1/x] the field of all meromorphic series at 0 either convergent or not and by K = C x [1/x] the subfield of the convergent { } ones. We consider lineare differential systems or equations with coefficients in K, i.e., with convergent meromorphic coefficients. The formal classification of linear differential systems or equations is the classification under K-equivalence (cf. Def 4.2.3 and Prop. 4.2.9). The meromorphic(2) classification is the classification under K-equivalence. e (2) We use the term meromorphic in the sense of convergent meromorphic. Otherwise, we specify formal meromorphic or simply formal. 72 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

In this section, we sketch the main theoretical results on the formal and the meromorphic classes of systems or equations. In the case of equations we also sketch the practical algorithms based on Newton polygons to compute the formal invariants.

4.3.1. The case of systems. — Denote by ′ the derivation with respect ′ dY to x, writing Y instead of dx , and consider an order one linear differential system (21) ∆Y Y ′ B(x)Y = 0 ≡ − with meromorphic coefficients (i.e., B(x) gℓ(n,K)). ∈ Recall (cf. Sect. 4.1) that a gauge transformation Z = TY changes the differential system Y ′ B(x) Y = 0 into the differential system Z′ TB(x) Z = 0 − − with TB = T ′ T −1 + TBT −1. When T (x) is meromorphic we denote T G = GL(n, C x [1/x]) the ma- ∈ { } trix TB(x) is also meromorphic. But the matrix TB(x) may also be convergent for some divergent T (x). We denote by G(B) the set of formal meromorphic gauge transformations T GL(n, C[[x]][1/x]) such that TB(x) is convergent. ∈ The set G(B) contains G. While G is a group,e G(B) is not. The meromorphic class of the system is its orbit under the gauge transformations in G while its formal classe is its (larger) orbit under those ineG(B). 4.3.1.1. Formal classification. — The formal classification of n-dimensional e meromorphic linear differential systems is performed by selecting, in each class, a system of a special form called a normal form. There exist algorithms to fully calculate a normal form of any given system (cf. end of Sect. 4.3.2.3). Theorem 4.3.1 (Formal fundamental solution and normal form)

1. To any system (21) : Y ′ = B(x) Y there is a formal fundamental solution (i.e., a matrix of n linearly independent formal solutions) of the form (x)= F (x) xL eQ(1/x) Y where J e ⊲ Q(1/x)= j=1 qj(1/x) Inj (assume the qj’s are distinct) is a diagonal matrix satisfying Q(0) = 0; its diagonal entries are polynomials in 1/x or in a L 1/p fractional power 1/t = 1/x of 1/x; the notation Inj stands for the identity 4.3. CLASSIFICATIONS 73

matrix of dimension nj. The smallest possible number p is called the degree of ramiﬁcation of the system, eQ(1/x) the irregular part of (x) and the q ’s Y j the determining polynomials. ⊲ L gℓ(n, C) is a constant matrix called the matrix of the exponents of ∈ formal monodromy. ⊲ F (x) GL(n, C[[x]][1/x]) is an invertible formal meromorphic matrix. ∈ e 2. The matrix (x) = xL eQ(1/x) is a (formal) fundamental solution of a Y0 system ′ Y = B0(x) Y ′ with polynomial coeﬃcients in x and 1/x. The system Y = B0(x) Y is formally equivalent to the initial system Y ′ = B(x) Y via the formal gauge trans- F formation F (x) (hence, B(x)= eB0(x)) and it is called a normal form of the given system Y ′ = B(x) Y . The fundamental matrix (x) is called a normal Y0 solution. e

A normal solution exhibits all formal invariants. However, the normal form and the normal solution are not unique: indeed, given P GL(n, C) any ∈ permutation matrix or any matrix commuting with Q(1/x), the matrix

−1 −1 P (x) P −1 = xPLP ePQ(1/x)P Y0 ′ P is also a normal solution associated with the normal form Y = B0(x)Y since a fundamental solution of the given system Y ′ = B(x)Y reads in the form (x) P −1 =(F (x) P −1) P (x) P −1 Y Y0 and F (x) P −1 still belongs to GL(n, C[[x]][1/x]). In the unramified case e (i.e., with ramification degree p = 1), a minimal full set of formal invariants is givene by the diagonal matrix Q(1/x) of the determining polynomials up to permutation and by the invariants of similarity of L (eigenvalues and size of the corresponding irreducible Jordan blocks). In the ramified case (i.e., with ramification degree p > 1) the situation is a little more intricate : given ′ a determining polynomial in the variable t′ = x1/p (i.e., with ramification degree p′) any element in its orbit under the action of the Galois group of ′ the ramification t′p = x is also a determining polynomial and any of them equally characterizes the orbit. In other words, a minimal set of invariants is well determined by one polynomial in each orbit jointly with the invariants of similarity of L. 74 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

′ Any normal form is meromorphically equivalent to Y = B0(x) Y . That’s why, sometimes, one generalizes the definition by calling normal form any ′ system meromorphically equivalent to Y = B0(x) Y . The theorem of formal classification was first proved in a weaker form, called Hukuhara-Turrittin Theorem, in which the given system Y ′ = B(x) Y is considered as a system in the ramified variable t = x1/p (p the degree of ramification of the system) allowing, thus, gauge transformations in a finite extension of the initial variable x (cf. [DMR07, p. 104, Thm. (4.2.1)] and also [Was76, HS99]). Stated as above it was first proved by W. Balser, W. Jurkat and D.A. Lutz [BJL79]. A simpler proof and an expression of the normal form in terms of rank reduced systems built on the minimal set of invariants can be found in [LR01]. Let us now state some definitions associated with the formal invariants. Choose a formal fundamental solution of System (21):

J (22) (x)= F (x)xL eQ(1/x) with Q = q (1/x)I and distinct q ’s Y j nj j Mj=1 e and the normal form Y ′ = B (x)Y with fundamental solution (x) = 0 Y0 xL eQ(1/x).

Definition 4.3.2 (Stokes arcs).— (i) Let q C[1/x] be a polynomial of degree k > 0 in the variable 1/x. We ∈ call Stokes arc associated with eq(1/x) (in short, with q) the closure of any arc of S1 of length π/k made of directions where eq(1/x) is flat. (ii) In the case of ramified polynomials q C[1/x1/p],p N∗, Stokes arcs ∈ ∈ can be defined similarly w.r.t. the variable t = x1/p on the corresponding p- sheet cover of S1. When the fractional degree of q is over 1/2 we call Stokes arcs of q their projection on S1 by the map t x = tp. Otherwise, the 7→ projections are onto S1 and one has to keep working with the variable t in the p-sheet cover . (iii) The Stokes arcs of a linear differential equation or system are the Stokes arcs associated with all its determining polynomials.

Example 4.3.3.— Suppose a determining polynomial of system (21) be given by q(1/x)= 1/x2/3. − 4.3. CLASSIFICATIONS 75

Then, the polynomials jq(1/x) and j2q(1/x) (where j3 = 1) are also determining polynomials of system (21). A fundamental solution of the system in the variable t = x1/3 con- 2 2 2 2 tains the three exponentials e−1/t , e−j/t and e−j /t to which correspond the six Stokes arcs deﬁned by π/4 arg(t) +π/4 mod π/3 and 3π/4 arg(t) 5π/4 mod π/3. By − ≤ ≤ ≤ ≤ projection of these six arcs on the circle S1 of directions in the variable x we obtain the two Stokes arcs deﬁned by 3π/4 arg(x) +3π/4 mod π, each one associated with the − ≤ ≤ three polynomials.

The matrix F (x) satisfies the homological system dF (23) = B(x) F FB (x). e dx − 0 which is a linear differential system in the entries of F and which admits the polynomials q q for j,ℓ = 1,...,J as determining polynomials and so, we ℓ − j can state: Proposition 4.3.4.— The Stokes arcs of the homological system (23) are the Stokes arcs associated with all polynomials q q for 1 j = ℓ J. ℓ − j ≤ 6 ≤ Split the matrix F (x) into column-blocks corresponding to the block- structure of Q (for j = 1,...,J, the matrix Fj(x) has nj columns): e F (x)= F (x) F (x) F (x) . 1 2 ···e J Definition 4.3.5 (Stokese arcse of Fej(x)).—eWe call Stokes arcs of Fj(x) the Stokes arcs associated with the polynomials qℓ qj for 1 ℓ J, ℓ = j. e − ≤ ≤ 6 e The Stokes arcs of the homological system are the Stokes arcs of all Fj(x).

Deﬁnition 4.3.6 (Levels, anti-Stokes directions) e We call (i) levels of system (21) the degrees of the determining polynomials q q ℓ − j for 1 j = ℓ J, of the homological system (23); ≤ 6 ≤ (ii) anti-Stokes direction associated with (24) (q q )(1/x)= a /xk 1+ o(1/x) = 0 ℓ − j − ℓ,j 6 any direction along which the exponential eqℓ−qj has maximal decay, i.e., , any direction θ = arg(a )/k mod 2π/k along which a /xk is real negative; ℓ,j − ℓ,j (iii) anti-Stokes directions of system (21) the anti-Stokes directions associated with all determining polynomials (q q )(1/x) = 0 of the homological ℓ − j 6 system (23); (iv) levels of F (x) the degrees of the polynomials q q for 1 ℓ J, ℓ = j; j ℓ− j ≤ ≤ 6 e 76 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

(v) anti-Stokes directions of Fj(x) the anti-Stokes directions associated with the polynomials qℓ qj for 1 ℓ J, ℓ = j. − ≤e ≤ 6 Observe that 0 is not a level since q = q for all ℓ = j and the polynomials ℓ 6 j 6 q contain no constant term. Notice, in the right hand side of (24), the minus sign which we would not introduce if we worked at inﬁnity. The anti-Stokes directions of a system are the middle points of the Stokes arcs of its homological system. The denomination “anti-Stokes directions” is not universal: sometimes, one calls such directions “Stokes directions” while to us, the Stokes directions are the oscillating lines of the exponentials eqℓ−qj .

It is worth to notice that it is always possible to permute the columns of a formal fundamental solution by writing it −1 −1 (x)= F (x) P xP LP eP Q(1/x)P Y with P the chosen permutation. It is also always possible to normalize a given e eigenvalue of L, say λ1, and a given determining polynomial, say q1, to zero by the change of variable Y x−λ1 e−q1 Y in the initial system (and at the same 7→ time, in its normal form). The Stokes arcs and the levels of F1(x) are then the Stokes arcs and the degrees of the determining polynomials qj themselves. e 4.3.1.2. Meromorphic classification. — The meromorphic classification proceeds differently than the formal one since it’s hopeless to exhibit (and, a fortiori, to calculate), in each meromorphic class, a system of a special form analogous to the normal form of the formal classification. Theoretically, the meromorphic classes are well identified as non-Abelian 1-cohomology classes. In practice, the meromorphic classes are identified via a finite number of matrices of a special form called Stokes matrices. Contrary to normal solutions, the Stokes matrices do not depend algebraically on the system; they are, in general, deeply transcendental with respect to the coefficients of the system. Some algorithms exist to calculate numerical approximations of the Stokes matrices in some “simple” situations but, yet, none is efficient in the very general case. Since the meromorphic classification refines the formal meromorphic one it is convenient, without any loss, to restrict the classification to a given formal ′ L Q(1/x) class with normal form Y = B0(x) Y and normal solution 0(x)= x e . ′ ′ Y Any system Y = B(x) Y in the formal class of Y = B0(x) Y satisfies, by F definition, a relation B(x)= eB0(x) for a convenient formal gauge transformation F (x) but such a gauge transformation F (x) is not unique in general: one

e e 4.3. CLASSIFICATIONS 77

−1 F1 F2 F F1 has e B0 = e B0 if and only if e2 e B0 = B0, i.e., if and only if there exists a ′ gauge transformation T which leaves invariant the normal form Y = B0(x) Y and such that F1(x)= F2(x)T (x). Notice that T (x) acts on F2(x) to the right. T The gauge transformations T for which B0 = B0 form a group. e e e Deﬁnition 4.3.7.— The group G0(B0) G(B0) of the gauge transforma- T ⊂ tions T for which B0 = B0 is called the group of isotropies or group of ′ invariance of the normal form Y = B0(x)Y . e

The group G0(B0) is in general small, even trivial, and it is easily determined in each particular case: it is made of all matrices T (x) such that there exists a matrix C GL(n, C) satisfying T (x) (x)= (x) C; this cor- ∈ Y0 Y0 responds to constant block-diagonal matrices C commuting with Q(3) and L −L such that x Cx is meromorphic. In the case when all diagonal terms qj in Q(1/x) are distinct the group G0(B0) is made of all invertible constant diagonal matrices; if, in addition, we ask for tangent-to-identity transformations then the group reduces to the identity.

Examples 4.3.8.— Denote by Ij the identity matrix of dimension j and by Jj the irreducible nilpotent upper Jordan block of dimension j. ⊲ Suppose the normal solution has the form

λ1I1⊕(λ2I3+J3) q1I1⊕q2I3 0(x)= x e Y where 0 < (λ1), (λ2) < 1 and where q1 = q2 are polynomials in 1/x. The invertible ℜ ℜ −1 6 matrices C such that 0(x) C 0(x) is a meromorphic transformation are those which Y Y commute both to eq1I1⊕q2I3 (this is a general fact) and to xλ1I1⊕(λ2I3+J3). One can check ∗ that this means that the matrix C has the form C = C1 C2 where C1 = cI1 with c C 2 ⊕ ∈ and C2 = c1I3 + c2J3 + c3J with c1,c2,c3 C and c1 = 0. All such constant matrices C 3 ∈ 6 form the group G0(B0).

Lj qj In ⊲ Suppose the normal solution has the form 0(x)= x e j with distinct qj ’s Y and matrices Lj = diag(λj,1,...,λj,n ) with integer coefficients λj,1,...,λj,n Z. j L j ∈ Then, C = Cj is any constant invertible block-diagonal matrix with Cj of dimension nj and the elements of G (B ) are the transformations of the form T (x) = xLj C x−Lj . L 0 0 j Their coefficients are polynomials in x and 1/x. L The meromorphic classes of formal gauge transformations of a system, ′ either a normal form or not, Y = B0(x) Y are, by definition, the elements of the quotient G G(B0) of all formal meromorphic gauge transformations of ′ \ Y = B0Y by the convergent ones to the left. The meromorphic classes of systems in the formale class of Y ′ = B Y are the quotient G G(B ) / G (B ) 0 \ 0 0 0

(3) If Q = L qj Inj then C = L Cj with matrices Cj of size nj . e 78 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

′ of the previous classifying set by the group G0(B0) of invariance of Y = B0 Y to the right (recall that the isotropies act on gauge transformations to the right, cf. supra). Since the group G0(B0) can always be made explicit it is sufficient to describe the classifying set G G(B ) of gauge transformations(4) \ 0 of the normal form. A first description of the meromorphice classes of gauge transformations was set up through a careful analysis of the Stokes phenomenon by Y. Sibuya [Sib77, Sib90] and by B. Malgrange [Mal79] (cf. Coms. 2.5.3). To state <0 1 their result we need to introduce the sheaf Λ (B0) over S of germs of flat ′ <0 isotropies of the normal form Y = B0(x) Y : a germ ϕ(x)inΛ (B0) is a germ (5) ϕ of GL(n, ) which is asymptotic to the identity and satisfies B0 = B0. The <0A sheaf Λ (B0) is a sheaf of non-commutative groups.

Theorem 4.3.9 (Malgrange-Sibuya).— The classifying set G G(B0) is 1 1 <0 \ isomorphic to the first (non Abelian) cohomology set H S ;Λ (B0) . e The map from G G(B ) into H1 S1;Λ<0(B ) is abstractly given by the \ 0 0 Main Asymptotic Existence Theorem (Cor. 4.4.4) while, way back, it is made explicit by means of Cauchy-Heinee integrals. Actually, meromorphic classes of gauge transformations can be given a simpler characterization as follows. ′ Let A be the set of anti-Stokes directions of the normal form Y = B0(x) Y <0 and denote by Stoα(B0) the subgroup of the stalk Λα (B0) made of all germs ′ of flat isotropies of Y = B0(x) Y having maximal decay at α. When α A the group Sto (B ) is trivial (no flat isotropy has maximal 6∈ α 0 decay but the identity). When α A the group Stoα(B0) can be given a ∈ L Q(1/x) linear representation as follows: given a normal solution 0(x) = x e J Y with Q(1/x)= j=1 qj(1/x) and distinct qj’s choose a determination α of α. Denote by (x) the function defined by (x) with that determination of Y0,αL Y0 the argument near the direction α. An element ϕα(x) of Stoα(B0) is a flat transformation such that ϕ (x) (x)= (x)(I + C ) α Y0,α Y0,α n α for a unique constant invertible matrix In + Cα. L Q(1/x) −Q(1/x) −L This implies that ϕα(x)= x e (In + Cα)e x with the given

(4) D.G. Babbitt and V.S. Varadarajan [BV89] call them meromorphic pairs (B , F ). 0 e (5) Flatness must be understood, here, in the multiplicative meaning of asymptotic to identity. 4.3. CLASSIFICATIONS 79

(ℓ,j) choice of the argument near α. Denote by Cα = [Cα ] the decomposition of Cα into blocks ﬁtting the structure of Q. Hence, the germ ϕα(x) reads

L (ℓ,j) (qℓ−qj )(1/x) −L ϕα(x)= x In + Cα e x .

k k An exponential eq(1/x) = e−a/x (1+o(1/|x |)) has maximal decay in a direction α S1 if and only if ae−ikα is real negative (k might be fractional). Hence, ∈ − q (1/x)−q (1/x) ϕα(x) is ﬂat in direction α if and only if, as soon as e ℓ j does not (ℓ,j) have maximal decay in direction α the corresponding block Cα of Cα vanishes. In particular, for j = ℓ, the exponential eqj −qℓ does not have maximal (j,j) q −q decay and the corresponding diagonal block Cα is zero; if e j ℓ has maxi- q −q (ℓ,j) mal decay in direction α then e j ℓ has not and thus, if a block Cα is not (j,ℓ) equal to zero the symmetric block Cα is necessarily zero. This implies that the matrix In + Cα is unipotent. Reciprocally, any constant unipotent matrix with the necessary blocks of zeros characterizes a unique element of Stoα(B0). Consequently, Stoα(B0) has a natural structure of a unipotent Lie group. The Malgrange-Sibuya Theorem has been improved by showing that in each 1-cohomology class there is a unique 1-cocycle of a special form called the Stokes cocycle which is constructible from any cocycle in its 1-cohomology class [LR94, Thm. II.2.1]; the uniqueness of the Stoles cocycle is further developed in [LR03].

Definition 4.3.10 (Stokes cocycle).— A Stokes cocycle is a 1-cocycle (ϕα)α∈A with the following properties: it is indexed by the set A of anti-Stokes directions and each component ϕα determines an element of Stoα(B0). The set of Stokes cocycles can be identified to the finite product Sto (B ) and we can state: α∈A0 α 0 TheoremQ 4.3.11 (Stokes cocycle).— The classifying set G G(B ) is isomorphic to the product Sto (B ) of 0 α∈A0 α 0 \ ′ the Stokes groups associated with a normal form Y = B0(x) Y . e Q From this theorem the classifying set inherits a natural structure of a unipotent Lie group. For applications of this property we refer to [LR94].

Let (ϕα)α∈A be a Stokes cocycle associated with a gauge transforma- ′ ′ tion F (x) from the normal form Y = B0(x) Y to a system Y = B(x) Y (hence, F (x) G(B )). Let (x) be a normal solution. Choose a determina- ∈ 0 Y0 tion αeof the argument for all α (it is usually understood that all α belong to a same intervale e ]2mπ, 2(m + 1)π]) and denote by (x) the normal solution Y0,α 80 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS with that choice of a determination of the argument. Finally, for all α A, ∈ let the matrix (In + Cα)α∈A represent ϕα with respect to these choices.

Deﬁnition 4.3.12.— The matrices (In + Cα)α∈A are called the Stokes matrices associated with the fundamental solution F (x) (x). Y0,α Like Stokes cocycles, Stokes matrices characterizee the meromorphic classes of gauge transformations: they form a full free set of meromorphic invariants. The Stokes cocycle and the Stokes matrices are connected to the theory of summation (Chap. 6) as follows. Suppose we are given a formal fundamental solution (x)= F (x) 0(x) at 0 and an anti-Stokes direction α A and denote + Y − Y ∈ by Fα (x) and Fα (x) the sums (k- or multisums) of F (x) to the left and to the right of the directione α. e Theorem 4.3.13.— The Stokes cocycle (ϕα)α∈A satisfy ϕ = F +(x)−1F −(x) for all α A. α α α ∈ The Stokes matrices (I +C ) A at 0 associated with F (x) (x) for a given n α α∈ Y0,α determination α of α satisfy e F −(x) (x)= F +(x) (x)(I + C ) for all α A. α Y0,α α Y0,α n α ∈

Formerly, one used to call Stokes matrices all matrices In + C satisfying a condition of the type

F (x) (x)= F (x) (x)(I + C) j Y0,α ℓ Y0,α n linking two overlapping asymptotic solutions, i.e., any matrix representing a germ of isotropy F (x)−1 F (x)= (x)(I + C) (x)−1, not necessarily a ℓ j Y0,α n Y0,α Stokes germ. This appeared to be not restrictive enough to easily characterize the meromorphic classes of systems or to exhibit good Galoisian properties: an example of a non-Galoisian “Stokes matrix” in the wide sense is given in [LR94, Sect. III.3.3.2]. Henceforward, we use the expression Stokes matrix in the restrictive sense of associated to a Stokes cocycle.

4.3.2. The case of equations. — The meromorphic and the formal equivalence of linear differential operators of order n were given in Definition 4.2.8 with a characterization in Proposition 4.2.9. Like for systems the formal class of an equation is made explicit from a formal fundamental solution which can be read as the first row of a formal 4.3. CLASSIFICATIONS 81 fundamental solution of its companion system. Each such solution takes the form φ(x) xλ eq(1/x) where the factors φ(x) are polynomials in ln(x) with formal series coefficients. Levels, Stokes arcs and anti-Stokes directions are defined similarly as for systems. The invariants are all the determining polynomials q(1/x) with multiplicities, the corresponding exponents λ and the degrees in ln(x) of each associated φ(x). The meromorphic classes in a given formal class are also characterized by (adequate) Stokes matrices. The formal invariants are much easier to determine for an equation than for a system. Below we sketch a procedure to follow for an equation. 4.3.2.1. Newton polygons. — Newton polygons are a very convenient tool to identify the formal invariants of a linear differential equation Dy = 0 at a singular point. By means of a change of variable any singular point can be moved to the origin 0. However, we state the definitions both at 0 and at infinity. Consider a linear differential operator dn dn−1 D = b + b + + b n dxn n−1 dxn−1 ··· 0 with coefficients bj that are either meromorphic series in x (for a study at x = 0) or in powers of 1/x (for a study at x = ). Temporarily, we ∞ do not need that the coefficients be convergent. The valuation of a power series b(x) = β xm at the origin is denoted by v (b) and defined m≥m0 m 0 as the smallest degree with respect to x of the non-zero monomials β xm P m of b; thus, v (b)= m when β = 0. The valuation of a power series 0 0 m0 6 b(1/x) = β /xm at infinity is denoted by v (b) and defined as m≥m1 m ∞ the highest degree with respect to x of a non-zero monomial β /xm of b; P m thus, v (b)= m when β = 0. When b is a polynomial in x, then v (b) ∞ − 1 m1 6 ∞ is the degree of b with respect to x.

Deﬁnition 4.3.14 (Newton polygons).— (i) Newton polygon at 0. — Suppose the coeﬃcients bj of D are formal or convergent meromorphic power series in x. With the operator D one associates in R+ R the set of marked points × PD = (j,v (b ) j) ; 0 j n . PD 0 j − ≤ ≤ 82 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The Newton polygon (D) of D at 0 is the upper envelop in R+ R of the N0 × various attaching lines of with non-negative slopes. PD (ii) Newton polygon at infinity. — Suppose the coefficients bj of D are formal or convergent meromorphic power series in 1/x. With the operator D one associates in R+ R the set of marked points × PD = (j,v (b ) j) ; 0 j n . PD ∞ j − ≤ ≤ + The Newton polygon ∞(D) of D at 0 is the lower envelop in R R of the N × various attaching lines of with non-positive slopes. PD Equivalently, we can say that the Newton polygon at 0 is the intersection of the closed upper half-planes limited by the various attaching lines of with PD non-negative slopes while the Newton polygon at infinity is the intersection of the closed lower half-planes limited by the various attaching lines of with PD non-positive slopes. One obtains the same Newton polygon when one enlarges the set of marked j points to any points (j, m j) corresponding to a non-zero monomial xm d − dxj in D or to the horizontal segments issuing from the points of backwards PD to the vertical axis.

m dj m Example 4.3.15.— Consider the operator D = x dxj . Since x is both a meromorphic series in x and in 1/x it makes sense to determine both its Newton polygon at 0 and at inﬁnity. There corresponds to D the unique marked point (j,m j) and the − corresponding Newton polygons are as shown on Fig. 1.

Figure 1

When D has polynomial coefficients one can define its Newton polygons both at 0 and at infinity. Definition 4.3.16 (Full Newton polygon).— Suppose D has polynomial coefficients. The full Newton polygon (D) is the intersection (D) N N0 ∩ (D) of the Newton polygons of D at 0 and at infinity. N∞ For simplicity and when there is no ambiguity, we denote by (D) anyone N of these Newton polygons. 4.3. CLASSIFICATIONS 83

Example 4.3.17.— Here below are the full Newton polygons of the Euler operator 2 d 3 d2 2 d = x dx + 1, its homogeneous variant 0 = x dx2 +(x +x) dx 1 and the hypergeometric E d d d E d − operator D3,1 = z z +4 z z +1 z 1 . dz − dz dz dz −

Figure 2

From now on, unless otherwise specified, we work at the origin 0, i.e., we suppose that D has formal or convergent meromorphic coefficients at 0. Proposition 4.3.18 (levels of D).— Suppose 0 is a singular point of D, i.e., at least one of the coefficients bj/bn has a pole at 0. (i) The levels of D at 0 are the positive slopes of (D). N0 (ii) The point 0 is regular singular for D if and only if the Newton polygon (D) has no non-zero slope. N0 Proposition 4.3.19.— Newton polygons satisfy the following properties. (i) Let D = xmD, m Z. The Newton polygon of D is the Newton m ∈ m polygon of D translated vertically by m.

(ii) Let D1 and D2 be two linear differential operators meromorphic at 0. Then, (D D )= (D )+ (D ). N0 1 2 N0 1 N0 2 Proof. — Assertion (i) is elementary. For a proof of (ii) we refer, for instance, to [DMR07, Lem. 1.4.1, p. 99]. As a consequence of (i), we may define the Newton polygon of an equation Dy = 0 as being the Newton polygon of D up to vertical translation. On the set C[[x]][1/x, d/dx] of linear differential operators at 0 it is convenient to introduce a weight (or 0-weight) w by setting dj dj dj w xk = k j and w xk = min w xk . dxj − dxj k,j dxj X In particular, w(x) = 1,w d = 1 and, to an operator D with weight dx − w(D)= w, the product x−wD has weight 0. At our convenience, given a differential equation Dy = 0, we can then assume that D has weight 0. 84 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Lemma 4.3.20.— Given j N and k Z, one has ∈ ∈ ′′ j j j k+1 d jk+j d j′ d x = x + c ′ ′′ x (c ′ ′′ C). dx dxj j ,j dxj′′ j ,j ∈ 1≤j′′

Observe that all monomials in the right hand side have weight jk and then, the whole expression in the left hand side has weight jk. The marked point associated with xjk+j dj/dxj is A =(j,jk). The marked points associated with the monomials in the sum are (j′,jk) with 1 j′ j 1, hence points ≤ ≤ − lying on the horizontal segment between A and the vertical coordinate axis.

Proof. — The formula in Lemma 4.3.20 is trivially true for j = 1. By Leibniz d k+1 k+1 d k rule we obtain the commutation law dx x = x dx +(k+1)x , from which it follows that d 2 d2 d xk+1 = x2k+2 +(k + 1)x2k+1 dx dx2 dx· Hence the formula for j = 2. The general case is similarly obtained by recurrence.

Proposition 4.3.21.— Given a diﬀerential operator D in the variable x denote by Dz the operator deduced from D by the change of variable x = 1/z. Then, the Newton polygons (D) and (D ) are symmetric with each other N0 N∞ z with respect to the horizontal coordinate axis.

d 2 d Proof. — One has dz = x dx . From Lemma 4.3.20 we know that we can − 2 d expand D in powers of the derivation δ = x dx with weight w(δ) = +1: D = c (x)δn + c (x)δn−1 + + c (x) n n−1 ··· 0 and the set of marked points is then given by (j,v (c )+ j) for 0 j n. PD 0 j ≤ ≤ Now, the operator Dz reads dn dn−1 D =( 1)nc (1/z) +( 1)n−1c (1/z) + + c (1/z) z − n dzn − n−1 dzn−1 ··· 0 and the associated marked points are (j,v c (1/z) j)=(j, v (c ) ∞ j − − 0 j − j). 4.3.2.2. Newton polygon and Borel transform. — We consider here the classical Borel transform (or 1-Borel transform) at 0 as deﬁned in Section 6.3.1 B below and we denote by ξ the variable in the Borel plane. We suppose that 4.3. CLASSIFICATIONS 85

D has polynomial coefficients in x and 1/x. As previously, we can expand D 2 d in powers of δ = x dx : D = c (x)δn + c (x)δn−1 + + c (x). n n−1 ··· 0 We assume that the coefficients cj are polynomials in 1/x. If this were not the −N case, we would replace D by x D with N the degree of the cj’s with respect to x. Let ∆ = (D) denote the operator deduced from D by Borel transform. B Since (δ)= ξ and 1 = d (cf. Sect. 6.3.1) the operator ∆ reads B B x dξ d d d ∆= c ξn + c ξn−1 + + c n dξ n−1 dξ ··· 0 dξ and is then a linear differential operator with polynomial coefficients. The fact that D had coefficients polynomial in 1/x is a key point here. In the general case, due to the fact that (fg)= (f) (g), the Borel transform of B B ∗B a linear differential operator is a convolution operator. The proposition below is a corollary of [Mal91b, Thm. (1.4)]. Proposition 4.3.22.— With normalization as above, the following two properties are equivalent: (i) the levels of D at 0 are 1; ≤ (ii) the levels of ∆ at infinity are 1. ≤ Proof. — Let v = min v (c ) 0 be the minimal valuation of the coefficients j 0 j ≤ of D at 0. This implies that all marked points associated with D at 0 are on the line issuing from (0,v) with slope 1 (Recall that δ has weight 1) or over and that at least one of them belongs to the line. As a consequence, all levels of D are 1 if and only if the point (n,v + n) of the line is a marked point, ≤ i.e., if and only if v0(cn)= v. To the other side, ∆ has degree n and order v. Similarly at 0, its − Newton polygon at infinity has no slope > 1 if and only if the monomial n d−v ξ dξ−v does exist in ∆. And indeed, this is precisely what the condition v0(cn)= v says. 4.3.2.3. Calculating the formal invariants. — We briefly sketch here how to calculate the formal invariants of a linear differential equation Dy = 0 with (formal) meromorphic coefficients at 0. Recall that the formal invariants at 0 of the equation are the determining polynomials q(1/x) with multiplicities, the exponents of formal monodromy λ and how many logarithms are associated with. 86 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

⊲ Indicial equation. — Suppose the Newton polygon (D) has a hori- N0 zontal side and consider the operator restricted to the marked points lying on that horizontal side. Up to a power of x to the left that operator reads dr dr−1 d D = γ xr + γ xr−1 + + γ x + γ , γ ,...,γ C. 0 r dxr r−1 dxr−1 ··· 1 dx 0 r 0 ∈ The indicial equation is the equation in the variable λ obtained by writing that xλ satisﬁes the equation D y = 0, i.e., denoting [λ] = λ(λ 1) ... (λ r + 1), 0 r − − the equation

γr [λ]r + γr−1 [λ]r−1 + ... + γ1 [λ]1 + γ0 = 0. λ Its roots λj (with multiplicities) are the exponents of factors x associated with no exponential. ⊲ k-characteristic equation. — Suppose the Newton polygon (D) has N0 a side with slope k and consider the differential operator restricted to the ′ s′′ marked points lying on that side with slope k. This operator reads xk D d k dxs′′ with ds ds−1 d D = c xs(k+1) + c x(s−1)(k+1) + + c xk+1 + c . k s dxs s−1 dxs−1 ··· 1 dx 0 k The k-characteristic equation is the equation obtained by writing that e−a/x satisfies the equation Dky = 0, i.e., c Xs + c Xs−1 + + c X + c = 0. s s−1 ··· 1 0 Its roots (counted with multiplicities) are the dominant coefficients a of the k exponentials e−a/x +··· times k. Differently said, they are equal to the domi- k nant coefficients ak in the derivatives of the exponentials e−a/x (1+0(1/x)). ⊲ Iterated characteristic equations. — Once one has determined the dominant coefficient a in the exponentials the next coefficients including the factor xλ attached to each exponential can be determined as follows. Select one root (−a/xk) a and consider the differential operator D1 = D deduced from D by the k k change of variable y = e−a/x Y (and simplifying by the factor e−a/x ). The ′ Newton polygon 0(D1) may have no slope k < k and no horizontal side; Nk in that case e−a/x is the exponential we look for and it comes factored with no xλ. It may have no slope k′

k term in the exponential e−a/x +··· and so on . . . until all exponentials and associated xλ are found. ⊲ Frobenius method. — When the indicial equations have multiple roots modulo Z there might exist logarithmic terms. To determine which terms appear with logarithms there exist a classical algorithm called Frobenius algorithm. Although the procedure is easy and natural from a theoretical viewpoint (it might be long and laborious in practice) we do not develop it here and we refer to the classical literature, for instance, [CL55, Sec 4.8]. When one knows all normal solutions, to complete them by formal series to get solutions of the initial equation Dy = 0 one proceeds by like powers identification. All these algorithms have been implemented in Maple packages such as Isolde or gfun. The case of systems is much more difficult to treat practically. There exists however algorithms to determine formal fundamental solutions. One can always apply the Cyclic vector algorithm (Sect. 4.1) and proceed as before. However, this way, there appear, in general, huge coefficients making the calculation heavy. It is then, in general, recommended to operate directly on the system itself. One method, which relies on Moser’s rank, was developed by M. Barkatou and his group (cf. [BCLR03] for a sketched algorithm and references). A variant was developed by M. Miyake [Miy11].

4.4. The Main Asymptotic Existence Theorem Consider a linear differential operator n dj D = b (x) j dxj Xj=1 with analytic coefficients at 0. The question here addressed is: is any formal solution of the equation Dy = 0 the asymptotic expansion of an asymptotic solution? A positive answer is given by the Main Asymptotic Existence The- orem (M.A.E.T.) either in Poincaréasymptotics or in Gevrey asymptotics. In the case of Poincaréasymptotics the theorem, precisely Cor. 4.4.2, is mostly due to Hukuhara and Turritin with a complete proof by Wasow [Was76]. An extension to Gevrey asymptotics is given by B. Malgrange in [Mal91a, Append. 1] and to non linear operators [RS89] by J.-P. Ramis and Y. Sibuya. 88 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The theorem roughly says that to a formal solution f of a diﬀerential equation (linear or non linear) there correspond actual solutions f asymptotic to f on various sectors. Given a direction, it is possible to determinee from the equation itself a minimal opening of the sector on which such an asymptotic solutione exists. However, these asymptotic solutions are, in general, neither unique nor given by explicit formulæ. Theorem 4.4.1 (Main Asymptotic Existence Theorem) The operator D acts linearly and surjectively on the sheaf <0 and on the A sheaves ≤−k for all k > 0. A In other words, the sequences <0 D <0 0 and ≤−k D ≤−k 0 for all k > 0 A −→A −→ A −→A −→ are exact sequences of sheaves of C-vector spaces. For the proof we refer to [Mal91a, Append 1; Thm. 1] where the theorem is stated and proved for all spaces

m Corollary 4.4.2.— Let f(x) = m≥0 amx be a power series solution of the diﬀerential equation Dy = 0. P e 1 (i) Given any direction θ S , there exists a sector = ′ (R) ∈ θ ]θ−δ,θ+δ [ and a function f ( ) such that ∈ A θ ⊲ Df(x) = 0 for all x (i.e., f is an analytic solution on ), ∈ θ θ ⊲ T θ f = f (i.e., f is asymptotic to f at 0 on θ).

(ii) If the series f(x) is Gevrey of order s then θ and f(x) can be chosen e e so that f(x) be s-Gevrey asymptotic to f(x) on θ. e Proof. — (i) The Borel-Ritt Theorem (cf. Thm. 2.4.1 (i)), provides for any e sector ′ containing the direction θ, a function g ( ′) with asymptotic ′ ∈ A expansion T ′ g = f on . Since T ′ is a morphism of differential algebras, <0 ′ T ′ Dg = DT ′ g = Df = 0. Hence, the function Dg is flat: Dg ( ). The ∈ A Main Asymptotic Existencee Theorem above applied to Dg in the direction θ provides a sector e ′ containing the direction θ and a function h <0( ) θ ⊂ ∈ A θ such that Dh = Dg. The function f = g h satisfies the required conditions − on θ. (ii) When the series f(x) is s-Gevrey the Borel-Ritt Theorem with Gevrey conditions (cf. Thm. 2.4.1.(ii)) provides a function g ( ′) over some sector ∈ As ′ containing the directione θ which is s-Gevrey asymptotic to f on ′. Its

e 4.4. THE MAIN ASYMPTOTIC EXISTENCE THEOREM 89 derivative Dg is asymptotic to Df(x) = 0 and, from Proposition 2.3.17, we can assert that Dg is k-exponentially ﬂat on ′. Hence, by the Main Asymptotic Existence Theorem, h belongs toe ≤−k and the conclusion follows as in the A previous case.

Since the proof relies on the Borel-Ritt Theorem it does not provide the uniqueness of the asymptotic solutions. The theorem does not make explicit the size of the sector θ. When the series f is convergent the sector θ can be chosen to be a full disc around 0 and f(x) to be the sum of the series. The opening of a possible sector can be verye diﬀerent depending on the series and on the chosen direction θ. The analysis of the Stokes phenomenon of the diﬀerential equation shows that in any direction one can choose a sector of opening at least π/k for k the highest level of the equation.

Comments 4.4.3 (On the examples of section 2.2.2)

⊲ Example 2.2.4. The Euler function is asymptotic to the Euler series on a sector of opening 3π and this sector is an asymptotic sector in any direction θ. The highest (and actually unique) level of the Euler equation is k = 1 and thus, the actual opening of 3π is larger than π/k = π. However, if we ask for a sector bisected by the direction θ the opening reduces to π in the direction θ = π. ⊲ Example 2.2.6. The hypergeometric function g(z) is asymptotic to the hypergeometric series g(z) on a sector of opening 4π while π/k = 2π (the unique level of the

hypergeometric equation D3,1y =0 is k =1/2). The anti-Stokes (and singular) directions are the directionse θ =0 mod 2π since the exponentials of a formal fundamental solution 1/2 are e±2 z . An asymptotic sector bisected by θ =0 has 2π as maximal opening. ⊲ In the previous two examples there exists only one singular direction and the possible asymptotic sectors are much larger than the announced minimal value. Actually, considering two neighboring singular directions θ<θ′ an asymptotic sector always exists with opening ]θ π/(2k),θ′ + π/(2k)[ for k the highest level of the equation. When − the singular directions are irregularly distributed the asymptotic sectors are “irregularly” wide.

Let ∆Y Y ′ B(x) Y = 0 be a linear diﬀerential system of order 1 ≡ − and dimension n with formal fundamental solution F (x) xL eQ(1/x). The Main Asymptotic Existence Theorem 4.4.1 and its Corollary 4.4.2 remain valid for systems in the following form. e 90 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Corollary 4.4.4.— The operator ∆ acts surjectively in <0 n and in n A ≤−k for all k > 0 and consequently, it satisﬁes in all direction θ S1 the A ∈ following properties: (i) There exists a sector θ = ]θ−ω,θ+ω′[(R) and an invertible matrix function F GL n, ( ) such that ∈ A θ ∆ F (x) xLeQ(1/x) = 0 for all x , ∈ θ

( T θ F = F (F is asymptotic to F at 0 on θ).

(ii) If an entry of F eis s-Gevrey then the correspondinge entry of F can be chosen to be s-Gevrey asymptotic on a convenient θ. e Proof. — This extension to differential systems follows from the fact that each entry of F (x) satisfies itself a linear differential equation with meromorphic ′ coefficients deduced from the homological system (23): F = BF FB0. e −

4.5. Infinitesimal neighborhoods of an irregular singular point While algebraic functions have moderate growth the form of formal solutions given above and the Main Asymptotic Existence Theorem show that solutions of linear differential equations at an irregular singular point may exhibit exponential growth or decay. Infinitesimal neighborhoods of algebraic geometry are then insufficient to discriminate between the various solutions. We define below infinitesimal neighborhoods for irregular singularities of solutions of differential equations as suggested by P. Deligne in a letter to J.-P. Ramis dated 7/01/1986 [DMR07]. This approach is developed, with an application to index theorems, in [LRP97].

4.5.1. Infinitesimal neighborhoods associated with exponential order. — We begin with a concept related only to the exponential order of growth or decay of the singularity under consideration. This concept will show up to be slightly too poor for a good characterization of k-summable series but it is a necessary step, at least for clarity. ⊲ Base space X. — From this viewpoint the infinitesimal neighborhood X of 0 in C is defined as a full copy of C compactified by the adjunction of a circle at infinity and endowed with a structural sheaf defined as below. For F obvious reasons we represent the infinitesimal neighborhood of 0 as a compact disc in place of the origin 0 in C. The “outside world” C∗ = C 0 is not \{ } 4.5. INFINITESIMAL NEIGHBORHOODS 91 affected by the construction and stays being endowed with the sheaf of germs of analytic functions. ⊲ Sheaf ≤k,k> 0. — Similar to the definition of k-exponentially flat A functions in Section 3.1.5 one says that a function f has exponential growth of order k on a sector if, for any proper subsector ′ ⋐ , there exist constants K and A> 0 such that the following estimate holds for all x ′: ∈ A f(x) K exp . | | ≤ x k | | The set of all functions with exponential growth of order k on is denoted ≤k by ( ) and one defines a sheaf ≤k over S1 of germs with exponential A A growth of order k (or, with k-exponential growth) in a similar way as ≤−k A (cf. Sect. 3.1.5). ⊲ Presheaf . — In view to define the sheaf it suffices to define the F F presheaf on a basis of open sets of X. We consider the following open sets F (cf. Fig. 3):

Figure 3. Basis of open sets in X

the discs D(0,k) for all k > 0, the (truncated) sectors I ]k′,k′′[= x = r eiθ ; θ I and 0

The restriction map between (truncated) sectors is the canonical restriction of functions and quotients. The restriction map from a disc D(0,k) to a sector is made possible by the isomorphism between C[[x]] and H0 S1; / ≤−k s As A following from the Borel-Ritt Theorem (cf. Seq. (19) and Cor. 3.2.8).

Notation 4.5.1.— From now on, a point x = k eiθ with k > 0 is also denoted by its polar coordinates (θ,k).

⊲ Sheaf . — The sheaf over X is the sheaf associated with the F F presheaf . F It is a sheaf of C-algebras. The stalk of at 0 is made of all Gevrey F0 F series. If useful, it could be extended to any series of C[[x]], the support of germs of non-Gevrey series having the point 0 as support. To deﬁne the { } stalk at the other points (θ,k) we introduce the sheaves

≤k− = lim ≤k−ε and ≤−k+ = lim ≤−(k+ε). −→ −→ A ε→0+ A A ε→0+ A

A germ f at θ belongs to ≤k− if there exist a sector in C∗ containing the A direction θ, an ε> 0, and constants K,C > 0 such that

Example 4.5.2 (Deﬁnition domain and support of exponentials)

An exponential exp a/xk + q(1/x) where q is a polynomial of degree less than k can − be seen as a section of the complement of the closed disc D(0,k) since it has exponential growth of order less than k′ for all k′ >k. 4.5. INFINITESIMAL NEIGHBORHOODS 93

On another hand, the exponential is ﬂat on the k open sectors arg(x)+ α/k < π/(2k) mod 2π/k where α denotes the argument of a. The exponential can then be continued on all of

these open sectors and is equal to 0 inside D(0,k). It cannot be continued any further. Its support is the complement of the open disc D(0,k) in its deﬁnition domain. The arcs on the circle of radius k limiting the sectors where the exponential is equal to zero are of length π/k. By analogy with the big points of algebraic geometry, their closure is called k-big points associated with the exponential exp a/xk +q(1/x) (or with the polynomial − a/xk + q(1/x)). On the picture are drawn two particular cases of the deﬁnition domain − (open shadowed part) of an exponential. The big points are in dotted lines. This example shows that the sheaf is in no way a coherent sheaf, hence its surname F of “wild analytic” sheaf.

The following properties of the sheaf are elementary and their proof is F left to the reader. Proposition 4.5.3.— The sheaf satisﬁes the following properties: F 1. The restriction 1 of to the circle centered at 0 with radius k F|S ×{k} F in X is isomorphic to the quotient sheaf ≤k−/ ≤−k+ over S1. A A 2. Sections over an open disc: H0 D(0,k); = lim C[[x]] = C[[x]] =: C[[x]] )C[[x]] . F ←− s+ε s+ε s+ s ε→0+ ε>0 \ 3. Sections over a closed disc: H0 D(0,k); = lim C[[x]] = C[[x]] =: C[[x]] (C[[x]] . F −→ s−ε s−ε s− s ε→0+ ε>0 [ 4.5.2. Inﬁnitesimal neighborhoods associated with exponential order and type. — As it follows from Proposition 4.5.3 the Gevrey space C[[x]] does not appear as a space of sections of over some disc or any other s F domain. To supply that gap we enrich the sheaf by taking into account F both exponential order and exponential type. 94 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

For a given k > 0 we define extensions Xk, k of X, as follows. F F ⊲ Base space Xk. — While, building X, we replaced the origin 0 by a copy of C compactified with a circle at infinity, we now replace the circle S1 k × { } of radius k in X by a copy Y k of C∗ = S1 ]0, [ compactified by the two × ∞ circles S1 0 and S1 . Precisely, we glue the lower boundary S1 0 ×{ } × {∞} ×{ } of Y k to the boundary of the disc D(0,k) and the upper boundary S1 × {∞} of Y k to the lower boundary of the complement X D(0,k). As topological \ spaces X and Xk are isomorphic to C.

Figure 4

We denote by (θ, k, ρ ) the polar coordinates of the points of Y k. A basis { } of open sets in Xk is given by open discs centered at 0 and truncated sectors like in X. k ⊲ Presheaf . — Given c > 0 we take into account the type c of F − exponentials of order k by introducing the subsheaf ≤k,c of ≤k and the + A A subsheaf ≤−k,c of ≤−k over S1 defined as follows. A −A A germ of ≤k,c at θ is a germ f ≤k which satisfies the condition: A ∈ Aθ there exist an open sector containing the direction θ, an ε > 0 and a constant K > 0 such that c ε f(x) K exp − for all x . ≤ x k ∈ | | + A germ of ≤− k,c at θ is a germ f ≤−k which satisfies the condi- A ∈ Aθ tion: there exist an open sector containing the direction θ, an ε > 0 and a constant K > 0 such that (c + ε) f(x) K exp for all x . ≤ − x k ∈ | | The space C[[x ]]s,C of series with fixed Gevrey order s and type C is the n subspace of C[[x]] made of the series n≥0 anx whose coefficients satisfy an P 4.5. INFINITESIMAL NEIGHBORHOODS 95 estimate of the form a K(n!)sCns for all n 0 and a convenient K > 0. | N | ≤ ≥ It is useful to introduce the spaces C[[x]]s,C+ = ε>0 C[[x]]s,C+ε. Thus, a series n a x belongs to C[[x]] + if for all ε> 0 there exists K > 0 such that n≥0 n s,C T P a K(n!)s(C + ε)ns for all n 0. | N | ≤ ≥ k In view to construct the sheaf k it suffices to define on a basis of F F open sets by setting: k inside X Y k, no change: = , \ F F  k k ′ ′′ 0 ≤k,c′− ≤−k,c′′+  inside Y : (I ] k,c , k,c [) = H I; / ,  F × { } { } A A  k k  across ∂Y : D(0, k,c ) = C[[x]]s,(1/c)+ , for 0

≤−k,c− T 0 + C[[x]] + 0, →A −→ As,(1/c) −−→ s,(1/c) → analog to the Borel-Ritt exact sequence (19) [LRP97, Sect. 1] . In this sequence the notation + stands for the following sheaf. A germ of + at θ As,C As,C is a germ f θ which satisﬁes the condition: there exist an open sector ∈A n containing the direction θ and a series n≥0 an x such that for all ε> 0 there is a constant K > 0 such that P N−1 f(x) a xn K (N!)s x N (C + ε)Ns on for all N N. − n ≤ | | ∈ n=0 X ⊲ Sheaf k. — The sheaf k is the sheaf over Xk associated with the k F F presheaf . It is a sheaf of C x -modules and no longer a sheaf of C-algebras F { } − − since the product of two functions of ≤k,c belongs to ≤k,(2c) and not − A A to ≤k,c in general. The stalk at a point (θ, k,c ) of Y k is given by A { } − ≤k / ≤−k if c = 0, Aθ Aθ k ≤k,c− ≤−k,c+ (θ,{k,c}) =  θ / θ if 0

Example 4.5.4 (Deﬁnition domain and support of exponentials)

An exponential exp a/xk + q(1/x) where q is a polynomial of degree less than k − and a = Aeiα, A> 0 is well defined in Y k for all (θ, k,ρ ) such that A cos(α kθ) <ρ. { } − − When cos(α kθ) 0 this means all points out of the “arch” ρ = A cos(α kθ). When − ≤ − − cos(α kθ) > 0 this leads to no constraint on ρ; moreover, the exponential is equal to 0 − inside the arch ρ = A cos(α kθ). − The k-big points associated with the exponential are now the closures of the arches ρ 0 where the exponential vanishes in Y k. − − In the example drawn in Figure 5 the definition domain of the exponential is the shadowed part of the infinitesimal neighborhood of 0. We indicated by “0” the open regions where the exponential vanishes. The big points are the closure of the arches (colored in orange-red in case of a colored copy) in which a small 0 is indicated.

i 1 Figure 5. Definition domain of exp x4 + q x (here, k = 4 and c = 1) We can now see the space C[[x]] as a space of sections of the sheaf k. s F Proposition 4.5.5.— Let D(0, k, 0 ) be the closure in Xk of the disc { } D(0,k) centered at 0 with radius k. Then, H0 D(0, k, 0 ); k = C[[x]] . { } F s Proof. — The equality follows from the fact that D(0, k, 0 )= c>0 D(0, k,c ) 0 k { } { } and that H D(0, k,c ); = C[[x]] + . { } F s,(1/c) T 4.5.3. More infinitesimal neighborhoods. — The previous construction can be repeated twice at levels k = k1 and k = k2 >k1 or finitely many times at levels k

Exercise 4.5.6. — Consider a linear diﬀerential equation dny dn−1y (25) Dy a (x) xn + a (x) xn−1 + + a (x) y = g(x) ≡ n dxn n−1 dxn−1 ··· 0 with formal series coeﬃcients a (x) C[[x]] and g(x) C[[x]][1/x]. We sup- j ∈ ∈ pose that g(x) is non-zero and we denote by p Z its valuation. ′ 2 d 1 ∈ Consider the operator D = g(x) dx g(x) D so that the equation dn+1y dny (26) D′ y b (x) xn+1 + b (x) xn + + b (x) y = 0 ≡ n+1 dxn+1 n dxn ··· 0 is the homogeneous form of equation (25). Denote by (D) and (D′) the Newton polygons at 0 of D and D′ respec- N N tively, ℓ and ℓ′ the lengths of their horizontal side and π(λ) and π′(λ) their indicial equations. (a) Prove that ℓ′ = ℓ + 1. (b) Prove that π′(λ)= C (λ p) π(λ) for a convenient constant C = 0. − 6 (c) When π(λ) has no integer root conclude that equation (25) admits a solution in C[[x]][1/x]. What happens when there exists r Z such ∈ that π(r) = 0?

Exercise 4.5.7.—

+∞ e−ξ/x (1) Check that the function F (x) = 0 ξ2+3ξ+2 dξ of exercise (2.5.4) satisﬁes the linear diﬀerential equation R (27) x4y′′′ + (2x3 + 3x2) y′ + 2y = x and explain the appearance of the exponential terms in the analytic continuation of F (x) over the Riemann surface of logarithms. Put equation (27) in homogeneous form

(28) D1 y = 0. Draw its Newton polygon at 0 and write its characteristic and indicial equations. Determine a fundamental set of formal solutions. Write down the companion system of equation (28), a normal form and a formal fundamental solution. 98 CHAPTER 4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Compute its Stokes matrix or matrices. (2) Consider the linear diﬀerential equation d d2 d (29) D y x2(x + 2) + 4(x + 1) x2 +(x2 + x) 1 y = 0 2 ≡ dx dx2 dx − where the factor to the right is the homogeneous Euler operator E0 (cf. Exa. 3.1.24). 2/x Show that y = e satisﬁes D2y = 0 and conclude that the equations D1y = 0 and D2y = 0 admit a same normal form (i.e., belong to the same formal class). Compute the Stokes matrices of D2y = 0 and conclude that the equations D1y = 0 and D2y = 0 do not belong to the same meromorphic class. CHAPTER 5

IRREGULARITY AND GEVREY INDEX THEOREMS FOR LINEAR DIFFERENTIAL OPERATORS

In this chapter, the results of the preceding sections are applied to prove index theorems for linear diﬀerential operators in the spaces of s-Gevrey series C[[x]]s as well as in the space C[[x]]∞ = C[[x]] of formal series and the space C[[x]] = C x of convergent series. The existence and the value of the 0 { } irregularity follow. We also sketch a method based on wild analytic continuation, i.e., continuation in the inﬁnitesimal neighborhood.

5.1. Introduction A linear map D : E E is said to have an index in E if it has finite −→ dimensional kernel ker(D,E) and cokernel coker(D,E). If so, the index is defined as being the number χ(D,E) = dim ker(D,E) dim coker(D,E). − An index is the Euler characteristic of the complex 0 0 E D E 0 0 · · · → −→ −→ −−−→ −→ −→ →··· where D is placed in degree 0 or even. It meets then all algebraic properties of Euler characteristics. In case coker(D,E) = 0 the index χ(D,E) gives the number of solutions of the equation Dy =0 in E. More generally, one says that a linear morphism L : E E′ between two vector spaces E and E′ has → an index if its kernel and its cokernel have finite dimension, the index being again the difference of these dimensions. From now on, we suppose that D is a linear differential operator dn dn−1 d D = b (x) + b (x) + + b (x) + b (x) n dxn n−1 dxn−1 ··· 1 dx 0 100 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS where the coefficients b (x) are convergent series at 0 C. The operator D is p ∈ a linear operator in any of the spaces C[[x]] for 0 s + and in any of s ≤ ≤ ∞ the quotients C[[x]] /C x . s { } The irregularity of D was first defined by B. Malgrange in [Mal74] as follows. Definition 5.1.1 (Irregularity).— The irregularity of D at 0 is the index of D seen as a linear operator in the quotient C[[x]]/C x . { } It was proved in [Mal74] that D has an index both in C[[x]] and in C x , { } the irregularity being then equal to χ(D, C[[x]]) χ(D, C x ). It was also − { } proved the relation coker(D, C[[x]]/C x ) = 0 which shows that the irregu- { } larity is the maximal number of divergent series solutions of the equations Dy = g(x) C x linearly independent modulo convergent ones. These in- ∈ { } dices were computed in terms of the coefficients bp(x) of D. The calculation of χ(D, C[[x]]) is elementary calculus (cf. Prop. 5.2.5 (i) below). The calculation of χ(D, C x ) follows from an adequate application of Ascoli’s Theorem. { } By a similar analytical method, based on direct or projective limits of Banach spaces and compact perturbations of operators, J.-P. Ramis [Ram84] extended these indices to a large family of Gevrey series spaces: the Gevrey spaces C[[x]]s as introduced above but also the Gevrey-Beurling spaces C[[x]] = lim C[[x]] = C[[x]] (s) ←− s,C s,C C>0 C>0 where \

C[[x]] = a xn C[[x]]; K > 0 such that a K(n!)sCns for all n s,C n ∈ ∃ | n| ≤ n nX≥0 o and the spaces C[[x]]s,C+ = ε>0 C[[x]]s,C+ε and C[[x]]s,C− = ε>0 C[[x]]s,C−ε. T S B. Malgrange [Mal74] and J.-P. Ramis [Ram84] computed also the indices of D acting on the fraction ﬁelds of formal, convergent or Gevrey series. They proved that these indices diﬀer by χ(D, C[[x]]) from those of D acting − in C[[x]], C x or C[[x]] for 0

A diﬀerential operator has no index in the spaces C[[x]]s,C themselves in general. A counter-example (cf. [LRP97, p. 1420]) is given by the Euler 5.1. INTRODUCTION 101

operator in C[[x]]1,1 as we prove below.

Proposition 5.1.2.— The Euler operator d = x2 1: C[[x]] C[[x]] E dx − 1,1 −→ 1,1 has no index when acting in C[[x]] for, coker( , C[[x]] ) has inﬁnite di- 1,1 E 1,1 mension.

Proof. — Check first that acts in C[[x]] . Suppose a xn satisfy E 1,1 n≥0 n a Kn! for all n. Then, | n| ≤ P a xn = b xn E n n nX≥0 nX≥0 where bn = (n 1)an−1 an satisfy bn 2Kn! for all n and the series n − − | | ≤ n≥0 bn x belongs to C[[x]]1,1. Consider now the family of series of C[[x]]1,1 P g (x)= (n 1)! nα xn, 0 <α< 1. α − n>0 X The unique series c xn solution of (y)= g (x) is given by c = 0 and n≥0 n E α 0 for n> 0 by c = (n 1)! 1α + 2α + + nα . The coefficients c have an n P− − ··· n asymptotic behavior of the form 1 1 c = (n 1)!n1+α 1+ O(1/n) = n! nα 1+ O(1/n) n α + 1 − α + 1 with α> 0[Die80, p.119, Exer. 27 or p. 305, Formula (7.5.1)]. Consequently, n the series n≥0 cnx does not belong to C[[x]]1,1 and gα(x) does not belong to the range of . Any non trivial linear combination λ g (x) has the same P E j αj property. P To prove that coker( , C[[x]] ) has infinite dimension it suffices to prove E 1,1 that the gα’s are linearly independent. To this end, suppose that the gα’s satisfy a linear relation of the form a1gα1 + a2gα2 + + argαr = 0. This means α1 α2 αr ··· that a1n +a2n + +arn = 0 for all n> 0. Choose n0 = 1. Applying the ···2 r 6 relation for n = n0,n0,...,n0 provides the van der Monde system based on α1 α2 α3 αr (λ1 = n0 ,λ2 = n0 ,λ3 = n0 ,...,λr = n0 ): λ a + λ a + + λ a = 0 1 1 2 2 ··· r r λ2a + λ2a + + λ2a = 0  1 1 2 2 ··· r r  ···  λra + λra + + λra = 0 1 1 2 2 ··· r r   102 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS

to determine the coeﬃcients a1,a2,...,ar which are then all equal to 0. Hence the g ’s are linearly independent and coker( , C[[x]] ) has inﬁnite dimension. α E 1,1

5.2. Irregularity after Deligne-Malgrange and Gevrey index theorems The proofs given in this section are due to B. Malgrange [Mal74] and P. Deligne [DMR07]) in letter to B. Malgrange, dated 22 aoˆut 1977. Introduce the following notations: ⊲ is the sheaf over S1 of germs of solutions of D; V ⊲ ≤k the subsheaf of germs with exponential growth of order at most k; V ⊲ <0 the sheaf over S1 of ﬂat germs of solutions of D;(1) V ⊲ ≤−k the subsheaf of germs with exponential decay of order at least k. V The sheaf ≤k is a subsheaf of ≤k, the sheaf ≤−k a subsheaf of ≤−k V A V A and the sheaf <0 a subsheaf of <0. All these sheaves are sheaves of C-vector V A spaces. The dimensions of the stalks of <0 and ≤−k at θ S1 are denoted V V ∈ by N <0(θ) = dim <0 and N ≤−k(θ) = dim ≤−k. Vθ Vθ Lemma 5.2.1.— The sheaves <0 and ≤−k for all k > 0 are piecewise V V constant. The functions θ N <0(θ) and θ N ≤−k(θ) are lower semi- 7→ 7→ continuous with jumps occuring only when entering or exiting a Stokes arc of D.

Proof. — Let D′ be a normal form of D. We denote by ′, ′≤k,... the V V sheaves associated with D′ as , ≤k,... are associated with D. By the Main V V Asymptotic Existence Theorem (Thm. 4.4.2) the sheaves <0 and ≤−k for V V all k > 0 are isomorphic to ′<0 and ′≤−k respectively and it is suﬃcient to V V prove the lemma for D′ instead of D. The space of solutions of D′y = 0 is spanned by functions of the form h(x)eqj (1/x) where h(x) has moderate growth at x = 0 and is deﬁned on the full germ of universal cover of C∗ at 0. Such functions belong to <0 in a direction θ if and only if eqj (1/x) belongs to <0. V Aθ If qj(1/x) appears with multiplicity mj in a formal fundamental solution of Dy = 0 then the solutions of the form h(x)eqj (1/x) generate a constant sheaf isomorphic to Cmj over the interior of each Stokes arc generated by eqj (1/x)

(1) The notations A≤0 and V≤0, in the continuation of the exponential case A≤−k and V≤−k for k > 0, are usually saved for germs with moderate growth. 5.2. IRREGULARITY AFTER DELIGNE-MALGRANGE 103 and nothing else. The same result is valid for ≤−k by considering only the V exponentials eqj (1/x) of degree at least k.

Comments 5.2.2 (Sheaf of solutions of the Euler equation)

Here below are drawn the unique Stokes arc of the Euler equation (cf. Exa. 2.2.4) and the graph of the function θ N ≤−1(θ). In this case, N <0(θ)= N ≤−1(θ). 7→

Figure 1

Theorem 5.2.3 (Deligne-Malgrange).— Any linear differential operator D with analytic coefficients satisfies the following properties. H1 S1; <0 for s =+ , 1. ker D, C[[x]]s/C x V ∞ { } ≃ H1 S1; ≤−k for 0 0. V 2 Proof. — 1.–2. Consider first the case s = + . The long exact sequence ∞ of cohomology associated with the short exact sequence 0 <0 → A → A → / <0 0 reads A A → 0 H0 S1; H0 S1; / <0 H1 S1; <0 H1 S1; → ≃ A −→ A≃ A −→ A −→ A ց ւ C x C[[x]] 0 { } for, H0(S1; <0) = 0, H0(S1; / <0) is isomorphic to C[[x]] by the Borel- A A A Ritt Theorem (cf. Cor. 3.2.8) and the map H1(S1; <0) H1(S1; ) factors A → A through 0 by the Cauchy-Heine Theorem (cf. Cor. 3.2.14). Hence, H1(S1; <0) C[[x]]/C x . A ≃ { } 104 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS

The Main Asymptotic Existence Theorem in sheaf form (Thm. 4.4.1) provides the short exact sequence 0 <0 <0 D <0 0. The associated long → V → A −→A → exact sequence of cohomology reads 0 H1 S1; <0 H1 S1; <0 D H1 S1; <0 0. ≃ → V −→ A −→ ≃ A → C[[x]]/C x C[[x]]/C x { } { } Hence, ker(D, C[[x]]/C x ) H1 S1; <0 and coker(D, C[[x]]/C x ) 0. { } ≃ V { } ≃ The case when s< + is proved similarly from the short exact sequences ∞ 0 ≤−k T / ≤−k 0 and 0 ≤−k ≤−k D ≤−k 0 →A →As−→As A → →V →A −→A → using the Gevrey parts of the Borel-Ritt and the Cauchy-Heine Theorems. 3. Denote by α for ℓ Z/pZ, the boundary points of the Stokes arcs of ℓ ∈ D ordered cyclically on S1 and by i : α ֒ S1 and j : ]α ,α [ ֒ S1 ℓ { ℓ} → ℓ ℓ ℓ+1 → the canonical inclusions. Since S1 is a real variety of dimension 1 the Euler characteristic(2) of the sheaf <0 satisfy V χ( <0) = dim H0 S1; <0 dim H1 S1; <0 . V V − V Then, χ( <0)= dim H1 S1; <0 since dim H0 S1; <0 = 0 (there exists V − V V no flat analytic function and, a fortiori, no flat solution all around 0 but the null function) and we are left to estimate the Euler characteristic of the sheaf <0. V Consider the short exact sequence 0 j <0 <0 i <0 0 → ℓ!V −→ V −→ ℓ∗V → Mℓ Mℓ The additivity of Euler characteristics allows us to write χ( <0)= χ(j <0)+ χ(i <0). V ℓ!V ℓ∗V Xℓ Xℓ The space H0 S1; j <0 is 0 since ]α ,α [ is not a closed subset of S1 (a ℓ!V ℓ ℓ+1 germ at a point of the boundary is the null germ by definition and generates null germs in the neighborhood, hence all over ]α ,α [). The sheaf j <0 ℓ ℓ+1 ℓ!V is a constant sheaf in restriction to ]αℓ,αℓ+1[ and 0 outside. Hence, the space 1 1 <0 1 <0 <0 H S ; jℓ! H ]αℓ,αℓ+1[; jℓ! is isomorphic to the stalk of at V ′ ≃ V <0 <0 V any point αℓ of ]αℓ,αℓ+1[ and therefore, χ(jℓ! )= dim α′ for all ℓ. V − V ℓ

~~(2) The number χ(F)= P(−1)j dim Hj X; F is, by deﬁnition, the Euler characteristic of a sheaf F over a space X. 5.2. IRREGULARITY AFTER DELIGNE-MALGRANGE 105~~

The space H1 S1; i <0 is 0 since the support of i <0 has dimension 0 ℓ∗V ℓ∗V and the space H0 S1; i <0 is isomorphic to the stalk <0 of <0 at α . ℓ∗V Vαℓ V ℓ Thus, χ(i <0) = dim <0 for all ℓ. ℓ∗V Vαℓ The number dim <0 dim <0 is both the variation of N <0 at α and at αℓ α′ ℓ V − V ℓ α . Hence, the 1 in the formula χ(j <0)+ χ(i <0)= 1 var N <0 . ℓ+1 2 ℓ ℓ!V ℓ ℓ∗V 2 This ends the proof of point 3. P P 4. The extension of the previous proof to the sheaf ≤−k is straightfor- V ward.

Remark 5.2.4. — The half variation of N <0 around S1 is also the number of Stokes arcs of D counted with multiplicity (cf. Def. 4.3.2 and Exa. 4.3.3), or equivalently, the sum of the (possibly fractional) degrees of all of the exponentials of a formal fundamental solution of D. The half variation of N ≤−k around S1 is the number of Stokes arcs of D with level at least k and counted with multiplicity, or equivalently, the sum of the degrees of all of the exponentials of degree at least k.

Corollary 5.2.5.— Let 0 < k < k < < k < + be the slopes of the 1 2 ··· r ∞ Newton polygon of the linear diﬀerential operator D (i.e., the levels of D) and denote, as usually, sj = 1/kj for j = 1,...,r. The operator D has an index in all spaces C[[x]] for 0 s + with values s ≤ ≤ ∞ (i) χ D, C[[x]] = lower ordinate of the Newton polygon (D); − N χ D, C[[x]] ♯ Stokes arcs of any level , i.e., (ii) χ(D, C x ) = − { } lower ordinate of the vertical side in (D); − N χ D, C[[x]] ♯ Stokes arcs of level

p p Proof. — Denote by v(bp) the valuation of the coeﬃcient bp(x) of d /dx in D and by m = inf v(b ) p the lower ordinate of the Newton polygon (D) p − N of D. (i) Prove that D has an index in C[[x]], equal to m. − 106 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS

From the definition of m the valuation of b satisfies v(b ) p + m for all p p ≥ p, the equality being reached on a non-empty set of indices p. Hence, the p+m P coefficient bp reads bp(x)= αpx + Ap(x) with v(Ap) >p + m for all p and the constant coefficient α is non-zero for p . For r m, we have p ∈P ≥− dp b (x) xr = r(r 1) (r p + 1)α xr+m + higher order terms. p dxp − ··· − p Hence, r r+m Dx = βr+mx + Br+m(x) where v(B ) >r + m. The constant β = r(r 1) (r p + 1)α r+m r+m p∈P − ··· − p being a polynomial with respect to r is non-zero for r r large enough. P ≥ 0 Denote by the maximal ideal of C[[x]] (ideal generated by x). The previous M calculation states that, for r r , the operator D induces a morphism D : ≥ 0 r r+m. M →M r+m Prove that this morphism is an isomorphism. Let g(x) = gr+mx + r Gr+m(x) with v(Gr+m) > r + m be given. A series f(x) = frx + Fr(x) with v(Fr) > r satisfies the equation Df = g if and only if fr = gr+m/βr+m and DFr = Gr+m + Cr+m for an adequate formal series Cr+m with valuation v(Cr+m) > r + m. The same reasoning applied to this new equation proves that the next term in f(x) is also uniquely determined and so on by recurrence. This achieves the proof of the fact that D : r r+m is an isomorphism. M →M Now, consider the commutative diagram

0 r C[[x]] C[[x]]/ r 0 −−→ M −−−→ −−−→ M −−→ ∼ D r+m   r+m 0  C[[x]] C[[x]]/ 0 −−→My −−−→ y −−−→ yM −−→ The left vertical morphism has an index equal to 0. The spaces C[[x]]/ r M and C[[x]]/ r+m being of ﬁnite dimension equal to r and r + m respectively M the right vertical morphism has an index equal to m. We can conclude that − the morphism in the middle has also an index χ(D, C[[x]]) and, by additivity of Euler characteristics, this index satisﬁes 0+ χ(D, C[[x]]) ( m) = 0. − − − Hence, the result. (ii) To prove that D has an index in C x with the value given in the { } statement we consider the exact sequence

~~0 C x C[[x]] C[[x]]/C x 0. → { } −→ −→ { } → 5.2. IRREGULARITY AFTER DELIGNE-MALGRANGE 107~~

Since D has an index both in C[[x]] and in C[[x]]/C x (cf. Thm. 5.2.3) it has { } also an index in C x and the three indices satisfy the addition formula { } χ(D, C x ) χ(D, C[[x]]) + χ(D, C[[x]]/C x ) = 0. { } − { } Hence, the result. (iii) The same argument applied to the exact sequence 0 C x C[[x]] C[[x]] /C x 0 → { } −→ s −→ s { } → proves that D has an index in C[[x]]s and provides the value given in the statement by the addition formula of Euler characteristics. (iv) follows directly from (i) and (ii). Remarks 5.2.6. — The following remarks are straightforward.

~~Figure 2. Newton polygon and the curve s χ(D, C[[x]]s) →~~

⊲ The proof above shows that the cokernel of D : C[[x]] C[[x]] can be → generated by polynomials. ⊲ The irregularity of D is zero if and only if the Newton polygon has only an horizontal edge. This corresponds, by definition, to a regular singular point at 0. Hence, the irregularity is zero if and only if 0 is a regular singular point. ⊲ The function s χ(D, C[[x]] ) is piecewise constant, increasing and left 7→ s continuous. ⊲ The indices χ(D, C[[x]] ), 0 s + , are not themselves formal mero- s ≤ ≤ ∞ morphic invariants since, in a gauge transformation, the Newton polygon is translated vertically. However, the irregularity is a meromorphic invariant as well as any difference of indices χ(D, C[[x]] ), 0 s + . s ≤ ≤ ∞ A consequence of the previous index theorem is the Maillet-Ramis The- orem. Maillet’s Theorem asserts that the series solutions of a linear or non 108 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS linear differential equation are Gevrey of a certain order k. J.-P. Ramis made the theorem more precise in the linear case by proving that the possible k are the levels of the equation.

~~Theorem 5.2.7 (Maillet-Ramis Theorem).— A series f(x), solution of the diﬀerential equation Dy = 0 is either convergent or s-Gevrey where s can be chosen so that k = 1/s be one of the levels k < k~~

A series solutione f(x) is a formal solution of the form f(x)e0 coming with a determining polynomial qj = 0. The levels associated with f(x) are the degrees of the polynomialse q q = q for all ℓ = j. Theye are, then, the non ℓ − j ℓ 6 zero slopes of the Newton polygon (D) of D at 0. e N0 Proof. — Let 0 < s < + and denote k = 0, k = + and s = ∞ 0 ∞ ∞ j 1/k for j = 0,..., + . Since, for all s, coker(D, C[[x]] /C x ) = 0 the j ∞ s { } number of independent solutions of D in C[[x]] /C x is equal to the index s { } χ(D, C[[x]] /C x ) of D in C[[x]] /C x . It is then constant for s

~~Comments 5.2.8 (On the examples of Section 2.2.2)~~

~~⊲ The Euler operator = x2d/dx + 1 and its homogeneous variant E 2 3 d 2 d 0 = x +(x + x) 1 E dx2 dx − are singular irregular at 0. They have same indices and same irregularity as indicated on~~

~~Fig. 3. Moreover, χ( , C[[x]]s) = χ( , C[[x]]) for s 1 and χ( , C[[x]]s) = χ( , C x ) E E ≥ E E { } for s< 1.~~

~~The unique non-zero slope of the Newton polygons ( ) and ( 0) is equal to 1 N E N E while the Euler series is 1-Gevrey and s-Gevrey for no s< 1 (cf. Com. 2.3.3).~~

~~Figure 3 5.3. WILD ANALYTIC CONTINUATION: INDEX THEOREMS 109~~

⊲ The exponential integral function satisfies i(y) = 0 where i is the operator E E d2 d i = x +(x + 1) . E dx2 dx This operator is regular singular at 0 and irregular singular at infinity. One can check that the Newton polygon at 0 reduces to a horizontal slope. The picture below shows the New- ton polygon ( i) at infinity. Recall that ( i) is the symmetric, with respect horizontal N E N E axis, of the Newton polygon at 0 of the operator i after the change of variable z =1/x. E Hence the change of signs in the indices. The series Ei(x) is 1-Gevrey.

Figure 4 ⊲ The hypergeometric operator d d d d D3,1 = z z +4 z z +1 z 1 dz − dz dz dz − is irregular singular at infinity. Its Newton polygon at infinity has a slope 0 and a slope 1 . − 2 Its indices and its irregularity at infinity are indicated on figure below.

~~Figure 5~~

~~The unique non-zero slope of the Newton polygon (D3,1) is 1/2 at iniﬁnity N − (hence +1/2 at 0 after the change of variable x = 1/z) and we saw that the hypergeometric series g(x) is 2-Gevrey.~~

e 5.3. Wild analytic continuation and index theorems We sketch here another method to compute a larger variety of index theorems for D. For more details we refer to [LRP97]. In that paper, indices are computed for D acting on a variety of spaces including the spaces considered before. The idea is to see each functional space as the 0-cohomology group of the sheaves , k, and so on. . . on a convenient subset of the base space F F 110 CHAPTER 5. IRREGULARITY AND GEVREY INDEX THEOREMS

X,Xk and so on . . . The admissible subsets U that are considered are ﬁnite unions of truncated narrow sectors (for a technical reason, sectors are assumed to be closed on their lower boundary) and, possibly, of a small disc centered at 0. The index χ(D, C[[x]]) of D acting on the space of formal series C[[x]] is assumed to be known, for instance, from a calculation as before (Cor. 5.2.5 (i)). From the present viewpoint the situation is made more complicated by the fact that the base spaces are now varieties of real dimension 2. The cohomology groups H2(U; ),H2(U; k) and so on. . . are non-zero groups but the F F cohomology groups Hi for i 1 satisfy (cf. [LRP97, Thms. 2.1 and 4.2]) the ≥ following sharp property. Theorem 5.3.1.— For i 1, the linear maps ≥ D : Hi U; Hi U; F −→ F are isomorphisms for all admissible U. The same result is valid for replaced by k, k1,k2 and so on ... F F F The technique is as follows. For small discs and narrow sectors (this means small enough to contain no big point associated with D) the calculation is elementary and based on the isomorphism between the spaces of solutions for D and for a normal form D′ of D over such domains. For a union of narrow sectors or of a small disc and narrow sectors the calculation follows from the use of Mayer-Vietoris sequences as follows (cf. [LRP97, Lem. 3.5]). Lemma 5.3.2.— Let U = U U where U and U are either open or closed 1 ∪ 2 1 2 subsets of U and suppose U, U ,U and U U are admissible subsets. 1 2 1 ∩ 2 If D has an index in H0(U ; ), H0(U ; ) and in H0(U U ; ) then, 1 F 2 F 1 ∩ 2 F it has an index in H0(U; ) given by F χ D,H0(U; ) = χ D,H0(U ; ) + χ D,H0(U ; ) F 1 F 2 F χ D,H0(U U ; ) . − 1 ∩ 2 F The same result is valid with replaced by k, k1,k2 and so on ... F F F For a (non-exhaustive) list of indices which can be computed that way we refer to [LRP97]. Let us just mention that the list includes indices of D acting on k-summable series over any k-wide arc I (cf. Def. 6.1.2) or acting on multisummable series over any multi-arc (I1,I2,... ) (cf. Sect. 8.7.1). These indices are formal meromorphic invariants of D as long as U does not contain a small disc about 0. Otherwise, their diﬀerence with χ(D, C[[x]]) are formal meromorphic invariants. CHAPTER 6

~~FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

Given a power series f(x) at 0 we know from the Borel-Ritt Theorem that there are infinitely many functions asymptotic to f(x) on any given sector with an arbitrary opening.e However, when f(x) satisfies an equation, these asymptotic functions do not satisfy the same equatione in general. The Main Asymptotic Expansion Theorem fills in this gape on small enough sectors for series solutions of linear differential equations by asserting the existence of asymptotic solutions. However, the theorem does not guaranty uniqueness and consequently lets the situation under some indetermination. The aim of a theory of summation on a given germ of sector (there might be some constraints on the size and the position of the sector) is to associate with any series an asymptotic function uniquely determined in a way as much natural as possible. What natural means depends on the category we want to consider. There is no known operator of summation applying to the algebra of all power series at one time and no hope towards such a universal tool. For the theory to apply to series solutions of differential equations an eligible request is that the summation operator be a morphism of differential algebras from an algebra of power series (containing the series under consideration) into an algebra of asymptotic functions (containing the corresponding asymptotic solutions). Both algebras must be chosen carefully and correspondingly. To sum series solutions of a difference equation one should look for a summation operator being a difference morphism; to sum basic series, for a summation operator being a q-morphism and so on . . . The simplest example is given by the usual summation of convergent power series from the algebra of convergent series into the algebra of germs of analytic functions at 0. Such a summation operator is indeed a morphism of differential algebras. 112 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

This chapter deals with the simplest case of summability called k- summability which applies to some divergent series and it aims at being a detailled introduction to the subject. We present four approaches which show up to be equivalent characterizations of k-summable series. With each approach we discuss examples and we attach some applications fitting especially that viewpoint. We give extensive proofs for most of the results and we refer to the literature when the proofs are omitted. A good part of the chapter can be found in [Mal95] or, in [Bal94] and [Cos09] (mostly for the Borel-Laplace approach) and [LRP97] (for an approach through wild analytic continuation). More references can be found in these papers and books. These questions were already widely considered by Euler. They have been developed at the end of the XIXth and the beginning of the XXth Cen- tury by mathematicians such as Borel, Hardy and al. A cohomological viewpoint brought them an impulse in the late 1970’s and 1980’s mostly with the works of Y. Sibuya, B. Malgrange, J.-P. Ramis, J. Martinet, W. Balser and lately, C. Zhang for basic series, giving rise to the abstract notions of simple or multiple summability. An extension of Borel’s approach was almost simultaneously developed by J. Ecalle,´ B. Braaksma, G. Immink, . . . , giving rise to the theory of resurgence and integral formulæ applying to a variety of situations. In the 1980’s J.-P. Ramis and Y. Sibuya [RS89] (see also, [LR90]) an- swered negatively the Turrittin problem [Was76, p. 326] by showing that series solutions of linear differential equations might be k-summable for no value of the parameter k > 0. They showed however that they are all, at worst, multisummable. The levels kj entering the multisummability process are the levels of the equation (cf. Def. 4.3.6). In the case of series solutions of linear difference equations J. Ecalle´ noticed that some series are neither k- summable nor multisummable. He showed that one has to introduce a new concept named k+-summability (cf. [E93´ ], [Imm96]) for a simple level k as well as for multiple levels k’s.

6.1. First approach: Ramis k-summability The problem we address now is to determine under which conditions the Taylor map T : H0(I; ) C[[x]] s,I As −→ s which, with a section of over an arc I of S1 (or of its universal cover As Sˇ1 R), associates its s-Gevrey asymptotic expansion, could be inverted as a ≃ 6.1. FIRST APPROACH: RAMIS k-SUMMABILITY 113 morphism of differential C-algebras. The answer is far from being straightforward and requires some restrictions both on I and C[[x]]s. The first definition of k-summability we present here is based on constraints for the asymptotic conditions themselves (recall k = 1/s). It relies on the results of chapters 2 and 3.

~~Comment 6.1.1 (On the Euler function (Exa. 2.2.4))~~

Although the problem here addressed is independent of any equation, what can happen is well illustrated by the behavior of the solutions of the Euler equation. We saw (cf. Coms. 2.3.9, p. 21) that E(x) is 1-Gevrey asymptotic at 0 to the Euler 3π 3π − series E(x) on any sector I based on the arc I = ] 2 , 2 [. Denote by E (x) and + − E (x) the two branches of E(x) on the half-plane −π = x ; (x) < 0 ; these branches { ℜ } are thee respective analytic continuations of E−π+ε(x) and E+π−ε(x) as ε > 0 tends to 0. The functions E−(x) and E+(x) are distinct. Indeed, if they were equal, E(x) would be asymptotic to E(x) all around 0 and this would imply that E(x) be convergent. More precisely, by applying Cauchy’s Residue Theorem, one can check [LR90] that e e E+(x) E−(x)=2πi exp(1/x) − The functions E−(x) and E+(x) are both 1-Gevrey asymptotic to E(x) at 0 on the half-

~~plane −π, and indeed, exp(1/x) is 1-Gevrey asymptotic to 0 on −π. When the sector is narrow, that is, when is at most an opene half-plane, then, E(x)~~

provides always a 1-Gevrey asymptotic solution on . However, when −π, the two ⊆ solutions E−(x) and E+(x), and hence all the solutions, are 1-Gevrey asymptotic to E(x). Existence is guarantied, uniqueness fails. When the sector is wide, that is, when contains a closed half-plane, then, eithere

~~does not contain the closure −π of −π and f provides the unique 1-Gevrey asymptotic~~

solution on , or contains −π and there is no 1-Gevrey asymptotic solution on . Uniqueness is guaranteed , existence may fail. In conclusion, there is no good size for an open sector to guaranty both existence and uniqueness of s-Gevrey asymptotic solutions. We will see that this property remains valid for s-asymptotic functions, not necessarily solutions. Note also that the defect of uniqueness is an exponential function. More generally, ﬂatness for solutions of linear diﬀerential equations is always related to exponential functions.

~~It is convenient to introduce the following deﬁnition.~~

Deﬁnition 6.1.2 (k-wide arc or sector).— ⊲ An arc I (of S1 or of its universal cover Sˇ1) is said to be k-wide if it is bounded and either closed with opening I π or open with opening I > π . | | ≥ k | | k ⊲ A sector is said to be k-wide if it is based on a k-wide arc. 114 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

It follows from the Borel-Ritt Theorem (Thm. 2.4.1 (ii) and Cor. 2.4.4) that the Taylor map T : H0(I; ) C[[x]] s,I As −→ s is surjective when the arc I is open with length I π/k and a fortiori, | | ≤ when I is closed with length I < π/k. Schematically, we can write | | I open or closed but not k-wide = T surjective ⇒ s,I

Consider now the injectivity of Ts,I . The example of the Euler function (cf. comment 6.1.1) shows that the Taylor map Ts,I may be not injective, at least, when I is small. For all I, the kernel of T is the space H0(I; ≤−k). s,I A Indeed, the left exactness of the functor Γ(I; . )= H0(I; . ) applied to the short exact sequence T 0 ≤−k s C[[x]] 0 (19) →A −→ As −→ s → implies exactness for the sequence

~~T 0 H0(I; ≤−k) H0(I; ) s,I C[[x]] . → A −→ As −−−→ s A suﬃcient condition for Ts,I to be injective is given by Watson’s Lemma.~~

Theorem 6.1.3 (Watson’s Lemma).— Let be an open sector with opening = π/k and suppose that f ( ) satisﬁes a global estimate of expo- | | ∈ O nential type of order k on , i.e., there exist constants C > 0, A> 0 such that the following estimate holds for all x : ∈ A f(x) C exp | | ≤ − x k · | | Then, f is identically equal to 0 on .

Roughly speaking, the lemma says: “under a global estimate of exponential order k on , the function f is too ﬂat on a too wide sector to be possibly non 0”. For a proof, among the many possible references, quote [Mal95, p. 174, Lem. 1.2.3.3] or to [Bal00, p. 75 Prop. 11]. In terms of sheaves Watson’s Lemma translates as follows.

~~Corollary 6.1.4 (Watson’s Lemma).— The sections of ≤−k over any A k-wide arc I are all trivial and consequently, the Taylor map Ts,I is injective. 6.1. FIRST APPROACH: RAMIS k-SUMMABILITY 115~~

~~Schematically,~~

k-wide arc I = H0(I; ≤−k)=0 = T injective ⇒ A ⇒ s,I Proof. — It suﬃces to consider the case when I is compact. A section of ≤−k A on I is represented by a ﬁnite and consistent collection of f ≤−k (R ) j ∈ A j j where the sectors (R ) have radius R and cover the arc I. Let R = j j j min (R ). Then, the f ’s glue together into a function f ≤−k( (R)) where j j j ∈ A (R) denotes the sector (R)= (R ) x

⊲ Choose = x ; π/2 < arg(x) < 3π/2 . Then, the exponential function exp(1/x) { } (which appears in the Euler example) belongs to ≤−1( ). Although this function is not A zero that’s not contradictory with Watson’s Lemma. Indeed, denote θ = arg(x); the best global estimate for exp(1/x) on is sup exp(1/x) = sup exp(cos θ/ x )=1 x∈ | | π/2<θ<3π/2 | | since cos θ tends to 0 as θ tends to π/2. ± ⊲ On another hand, the exponential exp(1/x) satisﬁes Watson’s estimate on any proper subsector of . This shows that Watson’s Lemma is no more valid on a smaller sector; here, for k = 1 on a sector of opening less than π and for any k > 0, using an adequate exponential of order k, on a sector of opening less than π/k. ⊲ Euler series. — We can now achieve our comment 6.1.1 and show that when

is a sector containing −π there exists no function (solution or not solution of the Euler equation) being 1-Gevrey asymptotic to the Euler series E(x) on . Indeed, suppose =]α,β[ ]0,R[ with α < π/2 < 3π/2 < β and f(x) be 1-Gevrey asymptotic × to E(x) on . In restriction to ]α, 3π/2[, the function f(x) E+(xe) is 1-Gevrey asymptotic − to E(x) E(x) 0; hence, it is 1-exponentially ﬂat (cf. Prop. 2.3.17) on a 1-wide sector − ≡ ande we can conclude by Corollary 6.1.4 of Watson’s Lemma that f(x)=E+(x)on]α, 3π/2[. Symmetrically,e e f(x)=E−(x) on ]π/2,β[. Hence, the contradiction since E+ =E− 6 on ]π/2, 3π/2[.

The conditions on the arc I to insure either the injectivity or surjectivity of the Taylor map Ts,I are complementary and there is no intermediate condition insuring both injectivity and surjectivity. In such a situation a natural solution proposed by J.-P. Ramis in the early 80’s to get both injectivity and surjectivity consisted in choosing for I a k-wide arc and restricting the space C[[x]]s of s-Gevrey series into a smaller space. n Suppose we are given a power series f(x)= n≥0 anx at 0.

~~Deﬁnition 6.1.6 (Ramis k-summability)e .—P 116 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

⊲ k-summability on a k-wide arc I (recall s = 1/k). — The series f(x) is said to be k-summable on I if I is a k-wide arc and f belongs to the range of the Taylor map T , i.e., if there exists a section f H0(I; ) whiche is s,I ∈ As s-Gevrey asymptotic to f on the large enough arc I. e ⊲ k-summability in a direction arg(x)= θ. — The series f(x) is said to be k-summable in the directione θ if there exists a k-wide arc I bisected by θ on which f(x) is k-summable. e ⊲ k-sum. — The function f above, which is uniquely determined when it exists, is callede the k-sum of f(x) on I or in the direction θ and we denote it by f = (f) or f = (f). Sk,I Sk,θ ⊲ k-summability. — Thee series f(x) is said to be k-summable if it is k-summablee in all directionse but ﬁnitely many, called the singular directions. e Notation 6.1.7.— We denote by C x the set of all k-summable series { }{k,I} on I and by C x the set of all k-summable series in direction θ. { }{k,θ} Notice that C x = C x for I the closed arc bisected by θ with { }{k,θ} { }{k,I} length π/k.

~~Remark 6.1.8. — It follows from the deﬁnition that a series which is k- summable in all direction is necessarily convergent.~~

~~Comment 6.1.9 (On the examples of chapter 1)~~

From Sect. 2.2.2 we deduce: ⊲ The Euler series E(x) of Example 2.2.4 is 1-summable according to the deﬁnition above: precisely, it is 1-summable in all directions but the direction θ = π. ⊲ Since we have note yet proved that the hypergeometric series g(z) of Example 2.2.6 is 1 a 2-Gevrey asymptotic expansion we cannot conclude yet about its possible 2 -summability. ⊲ In Example 2.2.7, as for the Euler function, we can movee the line of integration + from R to the half-line dθ with argument θ and get an estimate of the same type as Estimate (10) as long as π/2 <θ<π/2 (we leave that point as an exercise). This shows − that the series h(z) is 1-summable in all directions π/2 <θ<π/2. − ⊲ Similarly, one can show that the series ℓ(z) of Example 2.2.8 is 1-summable in all directions π/2e<θ<π/2. − b The following proposition is straightforward.

~~Proposition 6.1.10.— With deﬁnitions as above, and especially s = 1/k, we can state: 6.1. FIRST APPROACH: RAMIS k-SUMMABILITY 117~~

(i) The sets C x of k-summable series on a k-wide arc I { }{k,I} and C x of k-summable series in direction θ are differential subal- { }{k,θ} gebras of the Gevrey series space C[[x]]s; (ii) For I a k-wide arc of S1, the Taylor map T Γ(I, ) s,I C x As −−−→ { }{k,I} is an isomorphism of differential C-algebras with inverse the summation map . Sk,I As in Chapter 2 (cf. Prop. 2.3.13 and Cor. 2.3.14) let us now observe the effect of a change of variable x = tr, r N∗. Let I = (α,β) be a k-wide arc. j ∈ j In accordance with the notation /r for sectors in Section 2.3.2, denote by I/r the arc j I/r = (α + 2jπ)/r, (β + 2jπ)/r ′ 0 ′′ ℓ ℓ so that when θ = arg(t) runs over I/r = I/r then θ = arg(ω t) runs over I/r r j and θ = arg(x = t ) runs over I. Observe that I/r is kr-wide. Proposition 6.1.11 (k-summability in an extension of the variable) The following two assertions are equivalent: (i) the series f(x) is k-summable on I with k-sum f(x); r r (ii) the series g(t)= f(t ) is kr-summable on I/r with kr-sum g(t)= f(t ). e Proof. — The equivalence is a direct consequence of Definition 6.1.6 of k- e e summability and of Proposition 2.3.13. Given a series g(t) recall (cf. Sect. 2.3.2) that r-rank reduction consists in replacing g(t) by the r series gj(x),j = 0,...,r defined by e r−1 j r e eg(t)= t gj(t ) Xj=0 and that the series g (x) are given,e for j =e 0,...,r 1, by the relations j − r−1 j r ℓ(r−j) ℓ e rt gj(t )= ω g(ω t). Xℓ=0 From Corollary 2.3.14 wee can state: e Corollary 6.1.12 (k-summability and rank reduction)

~~The following two properties are equivalent: 118 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

(i) for ℓ = 0,...,r 1 the series g(t) is k′-summable on Iℓ with k′- − /r sum g(t); ′ (ii) for j = 0,...,r 1 the r-rank reducede series gj(x) is k /r-summable ′ − on I with k /r-sums gj(x) deﬁned by the relation

r−1 e rxj/r g (x)= ωℓ(r−j) g(ωℓx1/r), x1/r I . j ∈ 0/r Xℓ=0 In particular, a series g(t) is k′-summable if and only if its associated r-rank reduced series are k′/r-summable. e With these results we may assume, without loss of generality, that k is small or large at convenience. In particular, we may assume that k > 1/2 so that closed arcs of length π/k are shorter than 2π and can be seen as arcs of S1.

6.2. Second approach: Ramis-Sibuya k-summability Due to the quite simple integral formula defining the Euler function f(x) we were able to prove, in accordance to Definition 6.1.6, that the Euler series E(x) is 1-summable in all directions but the direction θ = π. However, to check s-asymptoticity on k-wide arcs is not an easy task in general (we refer for instancee to our other examples in Sect 2.2.2) and it is worth to look for equivalent conditions in different form. In this section, we discuss an alternate definition of k-summability, stated in the early 80’ by J.-P. Ramis and Y. Sibuya, which is based on series seen as 0-cochains. In order to work on S1 we assume that k > 1/2. This assumption does not affect the generality of the purpose as explained at the end of the previous section.

6.2.1. Deﬁnition. — Let = I be a “good” covering of S1 (hence I { j}j∈Z/pZ a covering without 3-by-3 intersections; cf. Def. 3.2.9). Its connected intersec- • tions 2-by-2 are the arcs (I = I I )(1) and, given a sheaf over S1, j j ∩ j+1 F a 1-cocycle of with values in is well deﬁned by the data of functions • • I F ϕ (I ) for all j Z/pZ. j∈ F j ∈

(1) There is an ambiguity with the notations when p = 2. In that case, the intersection I ∩ I is • • 1 2 made of two arcs which we denote by I 1 and I 2. 6.2. SECOND APPROACH: RAMIS-SIBUYA k-SUMMABILITY 119

Theorem 6.2.1 (Ramis-Sibuya Theorem).— • • Suppose ϕ =(ϕ ) is a 1-cocycle of with values in ≤−k. j j∈Z/pZ I A Then, there exist 0-cochains (fj Γ(Ij; ))j∈Z/pZ of with cobound- • ∈ A I ary ϕ and any such 0-cochain (f ) takes actually its values in , j j∈Z/pZ As i.e., f Γ(I ; ) for all j (recall that s = 1/k). j ∈ j As

The theorem says in particular that, under the condition that all the diﬀer- ences f + f are k-exponentially ﬂat, all the f ’s are s-Gevrey asymptotic − j j+1 j to a same s-Gevrey formal series f(x).

e • • • Proof. — When ϕ is trivial (ϕj= 0 for all j) then ϕ is the coboundary of any analytic function. Conversely, given any 0-cochain (fj) which, by means of a reﬁnement if necessary, we can assume to be a 0-cochain over a good covering the condition that its coboundary is trivial, i.e., f + f = 0 for all j, − j j+1 implies that the functions fj glue together into an analytic function f. The function fj are, in particular, s-Gevrey asymptotic to f on Ij for any s> 0. • When ϕ is elementary (i.e., only one of its components is non zero; • Def. 3.2.10) a 0-cochain with values in and coboundary ϕ is given by the As Cauchy-Heine Theorem 2.5.2 (ii). The general case follows by additivity of cocycles. In all cases, there exists then a 0-cochain (fj)j∈Z/pZ with values • in s and coboundary ϕ. Let (gj)j∈Z/pZ be another 0-cochain of with A • I coboundary ϕ. Then, the 0-cochain (g f ) has a trivial coboundary j − j j∈Z/pZ and comes from an analytic function h: for all j Z/pZ, g = f + h and ∈ j j then, like f , the function g belongs to (I ). j j As j The Ramis-Sibuya Theorem admits the following corollary:

~~Corollary 6.2.2.— The natural injection ֒ induces an isomorphism As →A~~

~~H0(S1; / ≤−k) i H0(S1; / ≤−k) As A −→ A A and, consequently (cf. Cor. 3.1.27), the Taylor map induces an isomorphism~~

~~H0(S1; / ≤−k) C[[x]] . A A ≃ s 120 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

~~We can thus improve the characterization of s-Gevrey series given in Sec- tion 3.2.3 into a characterization free of Gevrey estimates:~~

(equivalence class of a) s-Gevrey series 1  0-cochain (fj) over S f(x)= a xn C[[x]]   with values in n ∈ s ⇐⇒  A n≥0   and coboundary (fj fℓ)j,ℓ∈J X − e with values in ≤−k   A  This equivalence is a subsequent improvement with respect to the setting in Section 3.2.3 since to check that the 0-cochain is asymptotic in the sense of Poincaréis usually much simpler than to check its s-Gevrey asymptotics. While in Section 3.2.3 it was sufficient to ask for the coboundary to be with values in <0 it is now essential that the coboundary took its values in ≤−k. A A Definition 6.2.3 (k-quasi-sum).— Given f(x) an s-Gevrey series, the element ϕ H0 S1; / ≤−k associated 0 ∈ A A with f(x) by the Taylor isomorphism of Corollary 6.2.2 is called the k-quasi- e sum of f(x). By extension, any 0-cochain (fj) representing ϕ0 is called a k-quasi-sume of f(x). e With thesee results k-summability can be equivalently defined as follows.

Deﬁnition 6.2.4 (Ramis-Sibuya k-summability) An s-Gevrey series f(x) is said to be k-summable on a k-wide arc I with k-sum f(x) H0(I; ) if, in restriction to I, its k-quasi-sum ϕ satisﬁes the ∈ A 0 condition e ≤−k ϕ0| (x)= f(x) mod . I A

Indeed, suppose ϕ0 satisfies the condition above and let the 0-cochain (fj) be a k-quasi-sum of f(x). We know by Corollary 6.2.2 of the Ramis- Sibuya Theorem that all components fj are s-Gevrey asymptotic to f(x). Hence, the same is true foref(x) on the k-wide arc I and f(x) fits Definition 6.1.6. Conversely, a k-sum f(x) of f(x) in the sense of Definition 6.1.6,e can be completed into a k-quasi-sum (fj) of f(x) using the Borel-Ritt Theorem 2.4.1(ii) and Proposition 2.3.17. Thus,e we can reformulated Definition 6.2.4 by saying: e The s-Gevrey series f(x) is k-summable on the k-wide arc I with k-sum f(x) if there exists a k-quasi-sum of f(x) containing f(x) as component. e e 6.2. SECOND APPROACH: RAMIS-SIBUYA k-SUMMABILITY 121

6.2.2. Applications to differential equations. — As before we consider a linear differential operator with analytic coefficients at 0: dn dn−1 D = b (x) + b (x) + + b (x) with b (x) 0. n dxn n−1 dxn−1 ··· 0 n 6≡

⊲ The Maillet-Ramis Theorem (Thm. 5.2.7) can be obtained as a consequence of the Ramis-Sibuya Theorem as follows. Recall its statement: Given f(x), solution of Dy = 0, then, either f(x) is convergent or f(x) is s-Gevrey and k = 1/s is one of the levels k

Theorem 6.2.5.— Let f(x) be a solution of Dy = 0 and suppose the equation has a unique level k associated with f(x) (cf. Def. 4.3.6). Then, f(x) is k-summable. e e e 122 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

Proof. — Let θ be a direction which is not anti-Stokes for the equation Dy = 0 and Iθ be a k-wide arc centered at θ and containing no Stokes arc. Such a k-wide arc exists since, under the assumption of the unique level k, the Stokes arcs of Dy = 0 are the closed arcs of length π/k centered at the anti-Stokes directions. As in the proof of the Maillet-Ramis Theorem above, consider a 0-cochain (fj(x))j∈Z/pZ associated with f(x) made of asymptotic solutions. If we prove that, restricted to I its coboundary ( f +f ) is cohomologous to trivial via θ − j j+1 flat (hence, k-exponentially flat)e solutions then, the 0-cochain is cohomologous to a k-quasi sum of f(x) on Iθ. This is proved in Lemma 6.2.6 below. Hence, the k-summability of the series f(x) in all directions but, possibly, the finitely many anti-Stokes directions.e e Lemma 6.2.6.— Let <0 denote the sheaf of germs of flat solutions of the V equation Dy = 0 and suppose the arc I contains no Stokes arc. Then,

H1(I; <0) = 0. V ′ Proof. — Let <0 denote the sheaf of flat solutions of a normal equation V ′<0 D0y = 0 associated with Dy = 0. The property is easily proved for in- V ′ stead of <0. Indeed, by linearity, it is sufficient to consider the case when <0 V V is of dimension at most 1 (there is only one non-zero determining polynomial q) and when the covering is an elementary good covering of I, say =(I1,I2). A • ′ I non-zero 1-cocycle ϕ (x) Γ(I I ; <0) is of the form m(x)e±q(1/x) where ∈ 1 ∩ 2 V m(x) has moderate growth (precisely, is a linear combination of products of powers xλ and logarithm ln(x)) and e±q(1/x) is flat over I I . Since I does 1 ∩ 2 not contains any Stokes arc associated with q the exponential function e±q(1/x) is flat on at least one of the two open sets I1,I2. Suppose, for instance, it is • 0 ′<0 flat on I2. Then, ϕ can be continued into a flat function φ2 H (I2; ) • ∈ V and the 1-cocycle ϕ is the coboundary of the 0-cochain (f1 = 0,f2 = φ2) with ′ values in <0. V Now, the result is also true for <0 since, as a consequence of the Main V ′ Asymptotic Existence Theorem, the sheaves <0 and <0 are isomorphic. V V

~~Comments 6.2.7 (On Examples 2.2.4, 2.2.5 and 2.2.6)~~

~~Theorem 6.2.5 applied to these examples yields the following results: 6.2. SECOND APPROACH: RAMIS-SIBUYA k-SUMMABILITY 123~~

⊲ The Newton polygon ( ) at the origin 0 of the Euler equation (1) (cf. Exa. 2.2.4) N E and the Newton polygon at the origin 0 of the Euler equation 0y = 0 in homogeneous E form (cf. Exa. 3.1.24) are drawn below. The non-zero slopes reduce to a unique slope equal to 1. This implies that the exponentials in the formal solutions are all of degree 1. The fact that the horizontal length of the side of slope 1 is 1 means that there is only one such exponential (including multiplicity). The fact that ( 0) has one horizontal slope of length 1 means that there N E exists a one dimensional space of formal series solution of 0y = 0 (possibly factored by a E complex power of x; logarithms could also occur when the length is 2 and higher).

~~Figure 1. Numbers enclosed into brackets are the coeﬃcients to take into account in the indicial and the characteristic equations.~~

The Euler series E(x) is the unique, up to multiplication by a constant, series solution of the Euler equation in homogeneous form (Exa. 3.1.24). The exponent of the exponential is given by the characteristice equation associated with the slope 1, i.e., the equation r+1 = 2 R r0/x 1/x 0 with solution r0 = 1. Hence, the exponential e = e . The unique associated − anti-Stokes direction is θ = π. Theorem 6.2.5 allows us to assert what we were already able to prove directly on this very simple example: the Euler series E(x) is 1-summable in all direction but the direction θ = π. ⊲ The exponential integral Ei(z) has, ate inﬁnity, the same properties as the Euler function at 0 due to the formula Ei(z)= e−z E(1/z). ⊲ The Newton polygon at inﬁnity of the generalized hypergeometric equation D3,1y =0 Eq. (2.2.6) drawn below has a horizontal slope of length 1 and a slope 1/2 with horizontal length 2.

~~Figure 2 124 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

It follows that the equation has a one dimensional space of formal series solutions (space generated by the “hypergeometric” series g(z)); moreover, all exponentials are of degree 1/2. The characteristic equation associated with the slope 1/2 reads r2 1=0 −1/2 1/−2 with solutions r±0 = 1 and the exponentials aree exp( z ) = exp( 2z ). The ± ± ± anti-Stokes directions are the directions θ =2π mod 4π. (We need to go to the Riemann R surface of the logarithm since the slope is not an integer. After a ramification z = t2 we could stay in the plane C of the variable t: the anti-Stokes directions would become θ = π S1.) The indicial equation associated with the horizontal slope reads r +4=0 ∈ 4 with solution r0 = 4. Hence the factor 1/z in g(z). − Therefore, by Theorem 6.2.5, the series g(z) is 1/2-summable with respect to the variable z with singular directions θ =2π (mod.e 4π), which we had not proved earlier. e Theorem 6.2.5 holds for systems. Corollary 6.2.8.— Let dY/dx = B(x)Y be a differential system with a formal fundamental solution (x) = F (x) xL eQ(1/x) where Q(1/x) = J Y j=1 qj(1/x)Inj , the qj’s being distinct. Split the matrix F into column- blocks fitting the structure of Q: e L e F (x)= F (x) F (x) F (x) 1 2 ··· J (for j = 1,...,J, the matrix F (x) has n columns). Suppose the degrees of e je e j e the polynomials q q for ℓ = j are all equal to k. Then, the matrix F (x) ℓ − j 6 j (i.e., its entries) is k-summable.e e Recall that the matrix F (x) satisfies the homological system dF (23) = B(x) F FB (x) dex − 0 which admits the polynomials qℓ qj for j,ℓ = 1,...,J as determining poly- − dY nomials. B0(x) stands for the matrix of the normal form dx = B0(x)Y with fundamental solution (x)= xL eQ(1/x). Y0

~~6.3. Third approach: Borel-Laplace summation The third deﬁnition provides explicit k-sums in terms of k-Borel-Laplace integrals.~~

6.3.1. Deﬁnitions. — Due to its main role we ﬁrst make explicit the classical Borel-Laplace summation which corresponds to level k = 1. Actually, the general case k > 0 can be reduced to k = 1 by setting x = tk and taking t as a new variable. However, this introduces non integer powers in general with connected problems. We prefer to keep working with the initial variable x. 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 125

~~Deﬁnition 6.3.1 (Classical Borel and Laplace transforms)~~

(i) The Borel transform of a series f(x) = a xn is the power B n>0 n series e P n−1 f(ξ)= anξ /Γ n ; n>0 X b (ii) Given a direction θ, the Laplace transform of a function ϕ(ξ) in Lθ direction θ is deﬁned, when the integral exists, by

~~eiθ∞ f(x)= ϕ(ξ)e−ξ/x dξ. Z0~~

~~Although we do not need it in this chapter let us mention here that there exists a functional version of the Borel transform given, in each direction θ, by the integral~~

1 dx f(x) (ξ)= f(x)eξ/x Bθ 2πi x2 Zγθ where γ denotes the inverse (image by x 1/x) of a Hankel contour directed θ 7→ by the direction θ and oriented positively. (Let us observe that we need a contour that ends at 0 since the function is studied near the origin; if we worked at inﬁnity we would use a Hankel contour itself). Using Hankel’s formula for the gamma function we obtain (xn)= ξn−1/Γ(n) for all θ; hence, Bθ the coherence with the deﬁnition of the formal Borel transform. Similarly, (ξn−1) = Γ(n)xn. When there is no ambiguity we denote and instead Lθ B L of and . Bθ Lθ Observe that the formal Borel transform applies to series without constant term. With the constant 1 it would be natural to associate the Dirac distribution δ at 0. This is necessary in certain situations, for instance, when one needs to work with convolution algebras (δ is then a neutral element). For our purpose, this is unnecessary and we assume that our series have no constant term.

~~Here are some of the basic actions of the Borel transform. We denote by ϕ(ξ) both the Borel series of f(x) and its sum and by ψ both the Borel series~~

e 126 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY of g) and its sum. 1 d f(x) B ϕ(ξ) (assume f(x)/x has no constant term), e x −−→ dξ d x2 e f(x) B ξϕ(ξ), e dx −−→ ξ f(x)ge(x) B ϕ ψ(ξ)= ϕ(ξ η)ψ(η)dη. −−→ ∗ − Z0 and waye backe for the Laplace transform when it exists.

an n−1 Proposition 6.3.2.— Suppose the Borel series f(ξ) = n≥1 Γ(n) ξ converges and its sum ϕ(ξ) can be analytically continued to an infinite sec- P tor = ( ) with exponential growth at infinity:b there exist A, K > 0 θ1,θ2 ∞ such that ϕ(ξ) K exp A ξ on . ≤ | | Then, for all θ ]θ1,θ2 [, the Laplace integral ∈ eiθ∞ −ξ/x fθ(x)= ϕ(ξ)e dξ Z0 exists and is analytic on the open disc (A) with diameter (0, eiθ/A) and Dθ the various fθ glue together into an analytic function defined on θ θ(A) ′ ′ D and, especially, on a sector = θ(A) of opening greater θ1−π/2,θ2+π/2 ⊂ θ D S than π. S

~~Figure 3. Borel disc θ(A) D~~

Deﬁnition 6.3.3.— The disc (A) is called a Borel disc in direction θ. Dθ Proof.— Since ϕ(ξ)e−ξ/x K exp ( (eiθ/x) A) ξ the Laplace ≤ − ℜ iθ − | | integral fθ(x) exists and is analytic on the disc (e /x) > A, i.e., the open iθ ℜ disc θ(A) with diameter (0, e /A). D ′ ′′ ′′ ′ Consider two directions θ <θ in ]θ1,θ2[ such that, say, θ θ π/4 < −ξ/x − ≤ π/2 and apply Cauchy’s Theorem to ϕ(ξ)e along the boundary CR of 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 127 a sector of radius R limited by the lines θ = θ′ and θ = θ′′ and oriented counterclockwise. Then,

′ ′′ ′′ ℜeiθ Reiθ Reiθ + ϕ(ξ)e−ξ/x dξ = 0. iθ′ − Z0 ZRe Z0 However, denoting x = x eiω we can write | | iθ′′ Re ′′ −ξ/x θ iθ iθ ϕ(ξ)e dξ θ′ K exp ( (e /x) A) ξ ξ e dθ Reiθ′ ≤ − ℜ − | | | | Z ′′ = KR θ exp (cos(θ ω)/ x A)R Rdθ. θ′ − − | |− Choose θ′ <ω<θ′′. Then, θ Rω < π/4 for θ from θ′ to θ′′. The inequality | − | becomes ′′ Reiθ ϕ(ξ)e−ξ/x dξ K(θ′′ θ′)R exp 1/(√2 x ) A R Reiθ′ ≤ − − | | − Z and the integral tends to 0 as R tends to inﬁnity as soon as x < 1/(A√2). | | Consequently, the Laplace integrals in directions θ′ and θ′′ coincide on the domain x < 1/(A√2) and θ′ < arg(x) <θ′′ and they are, then, analytic {| | } continuations of each other. With this result we can set the following deﬁnition.

Definition 6.3.4 (Borel-Laplace summation).— A series f(x) = n n>0 anx is said to be Borel-Laplace summable in a direction θ0 if the following two conditions are satisfied: e P n−1 (i) The Borel transform f(ξ)= n>0 anξ /Γ n of f(x) is convergent, i.e., the series f(x) is 1-Gevrey. P (ii) The sum ϕ(ξ) of theb Borel series f(ξ) of f(x) cane be analytically e continued to a sector σ neighboring the direction θ0 with exponential growth of order 1. We still denote by ϕ its analytic continuation.b e When these conditions are satisfied, the Borel-Laplace sum of f(x) in direction θ0 is given by the Laplace integrals e eiθ∞ f (x)= ϕ(ξ)e−ξ/x dξ for θ σ θ ∈ Z0 gluing into an analytic function f(x) defined (at least) on a sector bisected by θ0 with opening larger than π. 128 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

This deﬁnition comes with explicit integral formulæ for the sum of the series. However, an explicit calculation of ϕ(ξ) in terms of classical functions is, in general, out of reach or even impossible. The domain of deﬁnition of ϕ(ξ) must contain a disc centered at 0 and a sector σ = σ]θ1,θ2[ neighboring the direction θ0 in the shape of a cham- pagne cork as below: the Borel-Laplace sum f(x) is analytic on the union (A) of the Borel discs with diameter (0, eiθ/A) for all direction θ in θ1<θ<θ2 θ D ′ ′ σ. The domain contains sectors ]θ′ ,θ′ [ for any θ >θ1 π/2 and θ <θ2 +π/2. S 1 2 1 − 2

~~Figure 4~~

These definitions can be extended to any level k > 0 as follows. Denote temporarily by or the classical Borel operators defined as above and B1 B1,θ generally, by or the k-Borel operators. Denote by the Laplace Bk Bk,θ L1,θ operator and generally, by the k-Laplace operator in direction θ. The Lk,θ operators of level k are transmuted from those of level one by means of rami- fications according to the following schemes. The k-Borel operators are defined by the following commutative diagram: B f(x) k,θ (f)(ξ)= ψ(ξk) −−−−→ Bk,θ

ρk ρ1/k 1/k B1,kθ x f(t ) ψ(τ) y −−−−−−−→  1 1/k τ/t 2 1 τ/xk k+1 where ψ(τ) = f(t )e dt/t = ′ f(x)e kdx/x , the path 2πi γkθ 2πi γθ ′ k γθ being deduced fromR the “Hankel” contourRγkθ by the ramiﬁcation t = x . The formal k-Borel transform is obtained by applying these formulæ to the monomials f(x)= xn. One obtains (xn)= ξn−k/Γ(n/k). In accordance Bk to the fact that the usual 1-Borel transform applies to series with valuation k 1, the k-Borel transform applies to series with valuation k k. 0 ≥ 0 ≥ The k-Laplace operators are deﬁned by the following commutative diagram: 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 129

~~′ Figure 5. The “Hankel” contour γkθ compared to γθ when k > 1~~

~~L ϕ(ξ) k,θ (ϕ)(x)= g(xk) −−−−→ Lk,θ~~

ρk ρ1/k 1/k L1,kθ x ϕ(τ ) g(t). y −−−−−−−→  ikθ e ∞ 1/k −τ/t where g(t)= 0 ϕ(τ )e dτ. We can thenR state the following deﬁnitions generalizing for any k > 0 the classical deﬁnitions of the Borel and the Laplace transforms stated above with k = 1.

~~Deﬁnition 6.3.5 (k-Borel and k-Laplace transforms)~~

(i) The (formal) k-Borel transform of a series f(x)= a xn with n≥k0 n valuation k0 k is the series ≥ e P ξn−k f(ξ)= an Γ n/k · nX≥k0 b (ii) The k-Borel transform of a function f(x) in a direction θ is deﬁned, when the integral exists, by

1 ξk/xk kdx k,θ(f)(ξ)= f(x)e k+1 B 2πi γ′ x · Z θ ′ with γθ a Hankel-type contour as above. (iii) The k-Laplace transform of a function ϕ(ξ) in a direction θ is deﬁned, when the integral exists, by

~~eiθ∞ k k f(x)= ϕ(ξ)e−ξ /x d(ξk). Zξ=0 130 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

Proposition 6.3.2 can be generalized to any level k > 0. The Borel discs must however be changed into Fatou petals (or Fatou ﬂowers) deﬁned in direction θ by conditions of the type eiθ/xk > A. ℜ

~~Figure 6. Fatou petal~~

Definition 6.3.6 (k-Borel-Laplace summation).— Let k k be an in- 0 ≥ teger. A series f(x) = a xn is said to be k-Borel-Laplace summable in a n≥k0 n direction θ if the following two conditions are satisfied: 0 P (i) Thee k-Borel transform f(ξ) = a ξn−k/Γ n/k of f(x) is con- n≥k0 n vergent, i.e., the series f(x) is 1/k-Gevrey. b P e (ii) The sum ϕ(ξ) of the Borel series of f(x) can be analytically continued e to a sector σ neighboring the direction θ0 with exponential growth of order k. We keep denoting by ϕ its analytic continuation.e If these conditions are satisfied, the k-Borel-Laplace sum of f(x) in direction θ0 is given by the k-Laplace integrals eiθ∞ e k k f (x)= ϕ(ξ)e−ξ /x d(ξk) for θ σ θ ∈ Zξ=0 gluing into an analytic function f(x) defined (at least) on a sector bisected by θ0 with opening larger than π/k. The natural question of the equivalence between k-summability and k- Borel-Laplace summability is studied in the next section (cf. Prop. 6.3.9).

6.3.2. Nevanlinna’s Theorem and summability. — We begin with the proof of Nevanlinna’s Theorem which solves the main step in the equivalence of k-summability and k-Borel-Laplace summability, precisely, the fact that k-summability implies k-Borel-Laplace summability. Assume we are given a direction θ issuing from 0 which by means of a rotation we assume to be θ = 0 and let us ﬁrst describe the curves and domains we will be concerned with. We consider two copies of C, one which we call the Laplace plane with coordinate x and the other one, called Borel plane, with coordinate ξ. 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 131

~~We ﬁx k > 0 and γ > 0 and we introduce two new copies of C with coordinates Z = 1/xk and ζ = ξk respectively.~~

⊲ In the x-plane we consider 1. The sector = x C ; x <γ and arg(x) < π/2k . 0 { ∈ | | | | } 2. For any ℓ> 0, the domain ( Fatou’s petal or Borel disc when k = 1) deﬁned by 1 π = x C ; >ℓk and arg(x) < . ℓ ∈ ℜ xk 2k n o

~~Figure 7~~

⊲ In the Z-plane, we consider the images 0 and ℓ of 0 and ℓ respectively, by the map Z = 1/xk. Hence, is the half-plane Z ; (Z) >ℓk and ℓ { ℜ } the half-plane Z ; (Z) > 0 but the half-disc Z 1/γk, (Z) > 0 . 0 { ℜ } {| | ≤ ℜ }

~~Figure 8~~

⊲ In the ζ-plane, for B > 0, we consider the domain Σ = D(0,Bk) Σ′ B ∪ B union of the open disc D(0,Bk) with center 0 and radius Bk and of the set ′ k k ΣB of points in C at a distance less than B of the line [B , + [. ∞ ′ ⊲ In the ξ-plane, we consider for B > 0, the domain σB = D(0,B) σB ∪′ union of the disc D(0,B) with center 0 and radius B and of the image σB of ′ 1/k ΣB by the map ξ = ζ for the choice of the principal determination of the kth-root. 132 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

~~Figure 9~~

~~Figure 10~~

~~Theorem 6.3.7 (Nevanlinna’s Theorem [Nev19, pp. 44–45])~~

Let k > 0. n Let f(x)= anx C[[x]] be a power series at 0 with valuation k0 k n≥k0 ∈ ≥ and let f(ξ)= an ξn−k denote its k-Borel transform. n≥k0 Γ(n/k) e P Suppose f(x) ( ) is asymptotic to f(x) and satisﬁes global k-Gevrey ∈P A 0 estimatesb on : there exist constants C,B > 0 such that for all x 0 ∈ 0 and N N∗ e ∈ N−1 N N/k x N (30) f(x) a xn C e−N/k | | − n ≤ k BN · n=k0 X Then, the k-Borel series f(ξ) is convergent and its sum ϕ(ξ) can be analytically continued to the domain σB with exponential growth of order k at inﬁnity: for any B = B ε 0 such that ε − ϕ(ξ) K exp A ξ k for all ξ σ . ≤ | | ∈ Bε Moreover, the functions f(x) andϕ(ξ) are k-Laplace and k-Borel transforms of each other: given ℓ>ℓ = inf ℓ ; f ( ) they satisfy 0 { ∈O ℓ } +∞ −ξk/xk k (31) f(x) = 0 ϕ(ξ)e d(ξ ) (k-Laplace transform of ϕ),

R 1 ξk/uk k (32) ϕ(ξ) = 2πi ℜ(1/uk)=ℓk f(u)e d(1/u ) for all ξ > 0. R Remark 6.3.8 (k-ﬁne summability). — One should observe that Condi- tion (30) is stronger than k-Gevrey asymptoticity of f(x) to the series f(x) on 0. Indeed, while Condition (30) is valid in restriction to any proper subsector e 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 133

of 0 (with same constants) implying thus k-astmptoticity on 0, conversely, the existence of estimates valid on any proper subsector of 0 does not imply the existence of constants B and C valid on all of 0. Note also that, due to Condition (31), when an analytic function f satisfying Condition (30) exists then it is unique. (Compare Watson’s Lemma (Thm. 6.1.3) and Proposition 2.3.17). This comforts the fact that Condition (30) is stronger than k-Gevrey asymptoticity. Recall the example of the Euler function E(x)(cf. Exa. 1 and Com. 2.3.9) that provides two functions (its two determinations) that are 1-Gevrey asymptotic to the Euler series E(x) on the half-plane (x) < 0. ℜ With these results we see that Condition (30) is adequate to guarantye the existence of a unique well defined sum of f(x) on 0 with similar properties as a k-sum. And indeed, this corresponds to a notion called k-fine summability in the bisecting direction of 0 which ise weaker than k-summability in the same direction. By this, we mean that a k-summable series in a direction θ0 is k-fine summable in direction θ0, the converse being false in general. For the case when k =1 we refer to [Sau], Sect. “The fine Borel-Laplace summation” where the author uses Formula (31) as definition.

Proof of Theorem 6.3.7 We can check that any monomial xn satisfies the theorem. Hence, we can assume that f(x)= a xn has valuation n≥k0 n k >k. 0 P Due to Condition (30) the series f(ξ) convergese at 0 with radius at least B (cf. Prop. 2.3.10). Its sum ϕ(ξ) defines then an analytic function on the disc ξ ℓ0 where 0 ℓ0 1/γ. 0 {ℜ } ≤ ≤ Choose ℓ>ℓ0 and set 1 φ(ζ)= F (U)eζU dU for all ζ > 0. 2πi k Zℓ +iR This formula makes sense. Indeed, in the new variables, Condition (30) becomes: for all N k0 and all Z Πℓk , the function F (Z) satisfies ≥ ∈ 0 N/k N−1 an N −N/k 1 (33) RN (Z) F (Z) C e | | ≡ − n=k0 Zn/k ≤ k (B|Z|1/k)N C′Γ(N/kP) 1 (using Stirling formula) ≤ (B′ |Z|1/k) N 134 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY for any B′ < B jointly with a convenient C′ > 0. In particular, there exist constants M0,M1 > 0 such that

M1 (34) F (Z) M0 for all Z Πℓk | | ≤ Z k0/k ≤ ∈ | | 1 k and since k0/k > 1 this implies that F (Z) belongs to L (ℓ + iR). Hence, its Fourier integral φ(ζ) exists and is continuous with respect to ζ R. ∈ We have to prove that: 1. The function F (Z) can be written in the form +∞ F (Z)= φ(ζ)e−Zζ dζ. Z0 2. The Borel series f(ξ) converges to φ(ξk) for 0 <ξ

~~1. Given Z Π k we enclose it in a domain Ω limited by the vertical line ∈ ℓ at ℓk and an arc of a circle centered at 0 with radius R as drawn in ﬁgure 11.~~

~~Figure 11~~

By Cauchy’s integral formula we can write 1 dU F (Z)= F (U) , 2πi Z U Z∂Ω − the boundary ∂Ω of Ω being oriented clockwise. From F (Z) M / Z k0/k | | ≤ 1 | | (estimate (34)) we deduce that the integral along the half-circle tends to zero 1 dU 1 as R tends to inﬁnity. Hence, F (Z) = 2πi ℓk+iR F (U) Z−U . Write Z−U = R 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 135

+∞ (U−Z)ζ 0 e dζ so that R 1 +∞ F (Z)= F (U) e(U−Z)ζ dζdU. 2πi k Zℓ +iR Z0 Fubini’s Theorem can be applied to the iterated integral since, using k again the estimate (34) we obtain F (U)e−(Z−U)ζ M1 e−(ℜ(Z)−ℓ )ζ ≤ |U|k0/k with (Z) ℓk > 0. Hence, the followed formulae ℜ − +∞ 1 F (Z)= φ(ζ)e−Zζ dζ with φ(ζ)= F (U)eUζ dζ. 2πi k Z0 Zℓ +iR Uζ Moreover, φ(ζ) is independent of ℓ>ℓ0 (apply Cauchy’s Theorem to F (U)e along a rectangle with vertical sides at (Z) = ℓk and (Z) = ℓ′k,ℓ′k = ℓk, ℜ ℜ 6 and let the horizontal sides go to inﬁnity). 2. From F (U)= N−1 a /U n/k+R (U) and φ(ζ)= 1 F (U)eUζ dζ n=k0 n N 2πi ℓk+iR for all ζ > 0 we can write P R

N−1 an n/k 1 Uζ φ(ζ)= ζ + φN (ζ) with φN (ζ)= RN (U)e dU. Γ(n/k) 2πi ℓk+iR nX=k0 Z By Condition (30), we have −N/k C N N/k e k dU φ (ζ) eℓ ζ N N N/k | | ≤ 2π k B ℓk+iR U Z | | while dU dτ dτ ℓN−k = < + . N/k N/k k /k ℓk+R U R √1+ τ 2 ≤ R √1+ τ 2 0 ∞ Z | | Z Z

Hence, there exists a constant C0 > 0 such that N N/k k 1 φ (ζ) C e−N/k+ℓ ζ | N | ≤ 0 k (ℓB)N · Take 0 <ζ 0 reaches its minimal value at ℓ = N 1/k 1 ℓN 1 k ζ1/k k N/k N/k N/k and y(ℓ1)= N e ζ . Choose n0 = n0(ζ) so large that n0 1/k 1 ℓ1 = >ℓ0 = inf ℓ ; f ( ) . k ζ1/k ∈O ℓ For N n0(ζ) and since φN (ζ) does not depend on ℓ>ℓ0, we can take ℓ = ℓ1. ≥ N/k Then, φ (ζ) satisﬁes φ (ζ) C ζ and tends to 0 as N tends to inﬁnity. N | N | ≤ 0 BN 136 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

Hence, φ(ξk)= an ξn = ϕ(ξ) for 0 <ξ 0. Given N N∗, let ∈ m N satisfy k(N + 1) < m k(N + 1) + 1. Write ∈ ≤ m an n/k 1 Uζ φ(ζ)= ζ + Rm+1(U)e dU Γ(n/k) 2πi ℓk+iR nX=k0 Z and look at the νth derivative of the integrand for 1 ν N. From (33) we ≤ ≤ can write ℓkζ ′′ ν Uζ ′ e ν−(m+1)/k C (35) Rm+1(U)U e C Γ (m + 1)/k U ≤ B′ m+1 | | ≤ U 1+1/k | | ′′ where the constant C is independent of ζ so long as ζ stays bounded. By Lebesgue’s Theorem we can then conclude that φ(ζ) can be derivated N times under the sign of integration for any ζ > 0. To estimate the N th derivative when ζ ζ Bk we write ≥ 0 ≥ ∂N φ(ζ)= J + I ∂ζN N N where m a n n n J = n 1 N + 1 ζn/k−N , N Γ(n/k) k k − ··· k − nX=k0 1 N Uζ IN = Rm+1(U)U e dU. 2πi k Zℓ +iR From the Gevrey Condition (30) or (33) (cf. Prop. 2.3.10) and the fact that Z 1/k 1/γ it follows that | | ≥ a C′ γ Γ((n + 1)/k) | n| + 1 Γ(n/k) ≤ B′n B′ Γ(n/k) for all n N∗, and since Γ((n+1)/k) behaves like n 1/k as n tends to inﬁnity ∈ Γ(n/k) k we can conclude that, for all B = B ε 0 ε − such that ′′ an C ∗ | | n for all n N . Γ(n/k) ≤ Bε ∈ 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 137

This implies that m 1/k n ′′ m m m −N ζ JN C 1 N + 1 ζ | | ≤ k k − ··· k − Bε nX=k0 m ζ1/k n ζm/k ζ1+1/k N k and, since n=k B m Bm k(N + 1 + 1/k) Nk+k+1 ζ if Bε ζ, 0 ε ≤ ε ≤ Bε ≤ that P ′′ 1+1/k kC 1 ζ k JN k+1 Γ N +3+ kN when Bε ζ. | | ≤ Bε k Bε ≤ From (35) and (33) we obtain

1 N ℓkζ IN Rm+1(U) U e dU | | ≤ 2π ℓk+iR | || | Z ℓkζ 1 e 1 1 dT C′Γ (m + 1)/k max , ′ m+1 2 1+1/k ≤ 2π B ℓ ℓ R √1+ T 2 k Z eℓ ζ C′′′Γ(N + 1 + 2/k) for a convenient C′′′ > 0 ≤ B′ kN ℓkζ ′′′ e C Γ(N + 1 + 2/k) kN ≤ Bε · Recall that k(N + 1) < m k(N + 1) + 1 so that 1 < m kN + 1 k 2. 1 1 ≤ − − ≤ Hence, the term max ℓ , ℓ2 and the power of the integrand. Adding these two estimates we see that, for all ε > 0, there exists a constant αε > 0 such that N ℓkζ ∂ φ e k (36) N (ζ) αεΓ(N + 3 + 2/k) kN for all ζ Bε . ∂ζ ≤ Bε ≥

Hence, the Taylor series of φ(ζ) at ζ , 0 1 ∂N φ (ζ )(ζ ζ )N , Γ(1 + N) ∂ζN 0 − 0 NX≥0 converges for ζ ζ < Bk. Making ε tend to 0, we can conclude that it | − 0| ε converges for ζ ζ < Bk and consequently, the Taylor series of φ(ζ) at ζ | − 0| 0 has a radius of convergence at least equal to Bk. To prove that this Taylor series converges to φ(ζ) write the Taylor- Lagrange formulas n−1 1 ∂pφ φ(ζ) = (ζ )(ζ ζ )p + ψ (ζ), Γ(1 + p) ∂ζp 0 − 0 n p=0 X 1 ∂nφ ψ (ζ) = ζ + θ(ζ ζ ) (ζ ζ )n, 0 <θ< 1. n Γ(1 + n) ∂ζn 0 − 0 − 0 138 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

k k k For ζ0 B and ζ Bε , then ζ0 + θ(ζ ζ0) > Bε and we can apply the ≥ n ≥ − estimate (36) to ∂ φ ζ + θ(ζ ζ ) so that ∂ζn 0 − 0 n k Γ(n + 3 + 2/k) ℓ max(ζ,ζ0) ζ ζ0 ψn(ζ) αε e | −kn | ≤ Γ(1 + n) Bε and tends to 0 as n tends to inﬁnity as soon as max(Bk,ζ Bk) <ζ<ζ +Bk. ε 0 − ε 0 ε Therefore, the sum of the Taylor series of φ at any ζ Bk coincides with 0 ≥ φ(ζ) on the interval max(Bk,ζ Bk) <ζ<ζ + Bk. This proves that φ(ζ) ε 0 − ε 0 ε admits an analytic continuation to Σ′ (ζ) >Bk . Bε ∩ {ℜ ε } Since the intervals ]0,Bk[ and ] max(Bk,ζ Bk),ζ + Bk[ for ζ = Bk, ε 0 − ε 0 ε 0 for instance, overlap this analytic continuation ﬁt the analytic continuation by ϕ(ζ1/k) on D(0,Bk) Σ′ (ζ) >Bk . ∩ Bε ∩ {ℜ ε } Letting now ε tend to 0 allows us to extend the analytic continuation of ′ φ(ζ)uptoΣB. Hence, the analytic continuation of ϕ(ξ) to the full domain σB.

4. Suppose 0 < Bε < B be given. Since ϕ(ξ) is analytic in the disc D(0,B) it is bounded in the smaller disc D(0,Bε). Consequently, φ(ζ) is k bounded in D(0,Bε ) and it suﬃces to prove the exponential estimate in the k k discs D(ζ0,Bε ) for ζ0 B . ≥ k The analytic continuation of φ to the disc D(ζ0,B ) is given by the Taylor series 1 ∂nφ φ(ζ)= (ζ )(ζ ζ )n. Γ(1 + n) ∂ζn 0 − 0 nX≥0 ∂nφ ′ Apply estimate (36) to ∂ζn (ζ0) with ε < ε. It follows that, on the disc k D(ζ0,Bε ), the function φ satisﬁes n k Γ(n + 3 + 2/k) ζ ζ0 ℓ ζ0 φ(ζ) αε′ | − | e | | ≤ Γ(1 + n) Bkn n≥0 ε′ X k n Γ(n + 3 + 2/k) Bε ℓkBk ℓkℜζ ′ ε αε k e e < + ≤ Γ(1 + n) Bε′ ∞ nX≥0 k (write ζ0 = (ζ0 (ζ)) + (ζ) and ζ0 (ζ) < Bε ). The estimate being −ℜ ℜ | −ℜ | k valid for all ℓ>ℓ0 we can conclude that there exist constants K > 0,A>ℓ0 such that φ(ζ) K eA|ζ| on Σ′ . | | ≤ Bε Hence the result.

~~Proposition 6.3.9.— k-Borel-Laplace summability in a given direction θ0 is equivalent to k-summability in the direction θ0. 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 139~~

Proof.— ⊲ k-Borel-Laplace summability implies k-summability. Suppose we are given a k-Borel-Laplace summable series f(x) = n n≥k anx in a direction θ0. This means that its k-Borel transform an ξn−k converges; its sum ϕ(ξ) can be analytically continuede to a Pn≥k Γ(n/k) sector σ containing the direction θ where ϕ(ξ) satisﬁes the following P ]θ1,θ2[ 0 inequality for some positive constants A and K: ϕ(ξ) K exp A ξk . ≤ | | By applying Cauchy’s Theorem one proves (proof left to the reader) that the k-Borel-Laplace integrals

∞eiβ −ξk/xk k fβ(x)= ϕ(ξ)e d(ξ ) Zξ=0 associated with the various directions β ]θ ,θ [ glue into a Borel-Laplace ∈ 1 2 sum f(x) deﬁned and analytic on the union of the Fatou ﬂowers

= x ; eiθ/xk > A D { ℜ } θ <θ<θ 1 [ 2 (recall that Fatou flowers are called Borel discs when k + 1). We must prove that f(x) is s-Gevrey asymptotic to f(x) on a sector bisected by θ ⊂ D 0 with opening larger than π/k (recall s = 1/k). Let β ]θ ,θ [ and ae sector bisected by β with opening ∈ 1 2 β ⊂ (π 2δ)/k < π/k be given. Prove that, under the hypothesis − ϕ(ξ) K exp A ξk for arg(ξ)= β ≤ | | ′ ′ there exist constants K ,A > 0 such that fβ satisfies N−1 N f (x) a xn K′Γ A′N x N β − n ≤ k | | n=k X for all x and N N∗ and that moreover, the constants K′,A′ are ∈ β ∈ independent of β while they depend on the size of β. We normalize the situation to the case when β = 0 by means of the rotation β both in the x- and the ξ-plane: the direction β becomes β∗ = 0 − and the variables x and ξ become x∗ = xe−iβ and ξ∗ = ξ e−iβ so that x∗ = x | | | | and ξ∗ = ξ ; we set f ∗(x∗)= f(x∗ eiβ) and ϕ∗(ξ∗)= ϕ(ξ∗ eiβ). We normalize | | | | to the case k = 1 by the change of variable ζ = ξ∗k (at the price of introducing ∗ ∗ non integer powers). The sector β has become a sector β bisected by θ = 0 140 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY with opening (π 2δ)/k. Setting φ(ζ)= ϕ(ζ1/k eiβ) we can write − +∞ ∗k f ∗(x∗)= φ(ζ)e−ζ/x dζ Z0 and φ satisfies φ(ζ) K eAζ for ζ > 0. ≤ Choose bk > 0 in the disc of convergence of φ and split the Laplace integral ∗ ∗ ∗ ∗ ∗ ∗ into f (x )= f1 (x )+ f2 (x ) with bk +∞ ∗ ∗ −ζ/x∗k ∗ ∗ −ζ/x∗k f1 (x )= φ(ζ)e dζ and f2 (x )= φ(ζ)e dζ. k Z0 Zb ∗ ∗ It follows from Lemma 2.4.2 that f1 (x ) is s-Gevrey asymptotic to the series ∗ ∗ ∗ iβ ∗ f (x )= f(x e ) on β with an estimate of the form N−1 e e N N (37) f ∗(x∗) a x∗n C∗ Γ A∗N x 1 − n ≤ 1 k 1 | | n=k X where A∗ = 1/(b sin(δ) 1/k) and C∗ = |an| bn. 1 | | 1 n≥k Γ(n/k) ∗ ∗ We must also prove that f2 (x ) is sP-Gevrey asymptotic to 0 with a global ∗ ∗ estimate on β. To this end, observe that f2 satisfies +∞ Abk ∗ ∗ A−ℜ(1/x∗k) ζ K e −bkℜ(1/x∗k) f2 (x ) K e dζ = ∗k e . | | ≤ k (1/x ) A Zb ℜ − When x∗ = x∗ eiθ belongs to ∗ then x∗k = x∗ k eikθ belongs to a sector | | β | | bisected by θ∗ = 0 with opening π 2δ. Then, cos(kθ) > sin(δ) and x∗k − satisfies (1/x∗k) > sin(δ)/ x k. Hence, ℜ | | Abk ∗ ∗ K e −(bk sin(δ))/|x|k f2 (x ) e . ≤ sin(δ)/ x k A | | − k The factor K eAb x k/(sin( δ) A x k) is bounded on ∗ and then, there exists | | − | | β constants A∗ = bk sin(δ) and C∗ > 0 such that

∗ k f ∗(x∗) C∗ e−A /|x| for x∗ ∗ . 2 ≤ ∈ β The constants depend on b and δ and not on β. From Proposition 2.3.17 we ∗ ∗ ∗ obtain that f2 (x ) is s-Gevrey asymptotic to 0 on β and that there exist ∗ ∗ constants C2 ,A2 > 0 depending on δ such that N (38) f ∗(x∗) C∗ Γ A∗N x N , 2 ≤ 2 k 2 | |

6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 141 the estimate being valid for all N N and x∗ ∗ (cf. proof of Prop. 2.3.17). ∈ ∈ β Hence, putting together (37) and (38), we can conclude that there exist constants C′,A′ > 0 such that for all N N∗ and x ∗ , the function f (x) ∈ ∈ β β satisﬁes N−1 N ′ f (x) a xn C′ Γ A N x N . β − n ≤ k | | n=k X Since the constants do not depend on the direction β the estimate (38) is valid ∗ for f(x) on the union = . Choosing δ < min(θ0 θ1,θ2 θ0) θ1<β<θ2 β − − implies that is a sector with opening larger than π/(2k) on both sides of the S direction θ0. Hence the result. ⊲ k-summability implies k-Borel-Laplace summability. n Suppose we are given a k-summable series f(x)= n≥k anx in a direction θ . This means that there exist an analytic function f(x) and constants 0 P B,C > 0 such that e

~~N−1 N f(x) a xn C Γ BN x N − n ≤ k | | n=k X on a sector = x; arg(x) θ < (π + 2δ)/2k and x 0 such that the Borel transform ϕ(ξ) is analytic and satisﬁes~~

k ϕ(ξ) K eA|ξ| ≤ on the domain σ′ equal to σ′ rotated by an angle β (cf. preamble of B,β B Nevanlinna’s Theorem). A rotation does not aﬀect the constants and thus, A and K being independent of the direction β, the estimate is valid on σ′ = σ′ which contains a champaign cork neighborhood of the |β−θ0|<δ/k B,β directionS θ0. Hence, the series f(x) is k-Borel-Laplace summable in direction θ0. e Comments 6.3.10 (On Examples 2.2.4, 2.2.7 and 2.2.8)

⊲ The Borel transform ϕ(ξ)=1/(ξ + 1) of the Euler series E(x) (example 2.2.4) has exponential growth of order one (and even less) in all direction. However, the sum of the Borel series cannot be continued up to inﬁnity in the direction θe= π due to the pole ξ = 1 of ϕ and indeed, we saw that the Euler series E(x) is 1-summable in all directions − but the direction θ = π. e 142 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

~~⊲ The Borel series h(ζ) of Example 2.2.7 has as sum the function ϕ(ζ)=1/(e−ζ 2) − which has exponential growth of order 1 in all directions. However, it has a line of~~

poles ζn = ln(2) + 2niπb , n Z. Hence, we can now conclude that the series h(z) − ∈ is 1-summable in all directions but θ = arg( ln(2) + 2niπ) for all n Z and their clo- − ∈ sure θ = π/2. In particular, it is not 1-summable in the sense of Definition 6.1.6 (point 4) ± which requires 1-summability in all directions but finitely many. This shows that solutions of difference equations, even when they are mild, can be not summable. −ζ ⊲ The Borel series ℓ(ζ) of Example 2.2.8 has as sum the function ϕ(ζ)= e−ζ+e −1 which is an entire function with exponential growth of order 1 in all directions (ζ) 0 ℜ ≥ and exponential growthb of no order in the directions (ζ) < 0. Hence, the series h(z) ℜ is 1-summable in all directions (z) > 0 and not 1-summable in the other directions. ℜ e 6.3.3. Tauberian Theorems. — The Tauberian Theorems we have in mind wish at comparing various k-sums of a given series in a given direction when several ones exist (cf. [Mal95] Théorème 2.4.2.2, [Bal94] Thms. 2.1 and 2.2). We begin with the following result.

Theorem 6.3.11.— Given numbers k1,k2 satisfying 0 < k1 < k2 deﬁne κ1 by 1/κ = 1/k 1/k . 1 1 − 2 Suppose we are given two closed arcs I1 and I1 with same middle point θ0 and respective length I = π/k and I = π/κ . Given a formal power | 1| 1 | 1| 1 series f(x) C[[x]] denote by g(ξ) = (fb)(ξ) its k -Borel transform ∈ Bk2 2 (cf. Def. 6.3.1). b e e The following two assertionsb are equivalent.

~~(i) The series f(x) is k1-summable on I1 with k1-sum f(x);~~

(ii) The series g(ξ) is κ1-summable on I1 and its κ1-sum g(ξ) can be e analytically continued to an unlimited open sector σ, containing I ]0, + [, 1× ∞ with exponential growthb of order k2 at inﬁnity.b Moreover, k (f)(ξ) = g(ξ) and k (g)(x) =bf(x) in directionb θ0 and B 2 L 2 neighboring directions.

Proof. — The theorem being true (and empty) for monomials we can assume that the series f(x)= a xn has valuation k >k . n≥k0 n 0 2 ⊲ Prove that (i) implies (ii). We proceed as in the proof of Nevanlinna’s e P Theorem. By assumption, the series f(x) has a k1-sum f(x) on the closed arc ′ I1, hence, on a larger open arc. Thus, there exists a closed arc I1 containing I1 in its interior, there exist r0 > 0e and constants A, C > 0 such that the 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 143 estimate N−1 (39) f(x) a xn CN N/k1 AN x N − n ≤ | | n=k0 X holds for all N N∗ and all x in the sector ′ = I′ ]0,r ]. ∈ 1 1× 0 For convenience, we normalize the Borel transform into the classi- Bk2 cal Borel transform of level 1. To this end, set Z = 1/xk2 , ζ = ξk2 and B1 R = 1/rk2 . In the coordinate Z, the sector ′ = I′ ]0,r ] is changed into 0 0 1 1× 0 a sector = J [R , + [ with opening larger than π (indeed, k > k ). 1 1 × 0 ∞ 2 1 The series f(x) becomes the series F (Z)= f(1/Z1/k2 ) and the function f(x) the function F (Z)= f(1/Z1/k2 ). In the coordinate ζ, the function g(ξ) becomes G(ζ)=e g(ζ1/k2 ). e e

Suppose ﬁrst that 1/k1 < 2/k2 (hence, k2π/(2k1) < π). Recall that, by assumption, we have 1/k2 < 1/k1 (hence, k2π/(2k1) > π/2). After performing a rotation to normalize the direction θ0 to 0 we get the following picture (Fig. 12) where J =[ ω ,ω ] with ω = k π/(2k )+ ε /, (suppose ε chosen 1 − 1 1 1 2 1 so small that ω < π). 1

~~Figure 12~~

~~Denote by g(ξ) = (f)(ξ) the k -Borel transform of f(x) in direc- Bk2,θ0 2 tion θ0 = 0. The Borel path to deﬁne G(ζ)= g(ζ1/k2 ) can be chosen as the boundary~~

∂ = ∂ ∂ ∂ 1 −1 ∪ 0 ∪ +1 of 1 where ∂0 denotes the part of the boundary which is a circular arc of radius R0 and ∂±1 the two straight lines of ∂ 1. Assume that ∂ 1 is oriented as in 144 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

Fig. 12. From Cauchy’s Theorem the path ∂ 1 can equivalently be deformed into its homothetic λ = λ λ λ where λ has radius R>R or into −1 ∪ 0 ∪ +1 0 0 a broken line ℓ = ℓ ℓ passing through a large enough α > 0 as shown in − ∪ + Fig. 12. Thus, γ being any of the Borel paths above, G(ζ) reads as 1 (40) G(ζ)= F (U)eζU dU. 2πi Zγ It suffices to prove that 1. the function G(ζ) is defined and holomorphic on the unlimited open sector Σ = J ]0, + [ where J =] ω, +ω[ with ω = k (π/(2κ )+ ε); × ∞ − 2 1 2. the function G(ζ) has exponential growth of order one at infinity on Σ and has F (Z) as Laplace transform; 3. there exist constants C′,A′ > 0 such that the following estimate holds for all N and all ζ Σ′ = J ′ ]0, 1/R [ where J ′ =[ ω′, +ω′], ω′ = ω k ε/2: ∈ × 0 − − 2

N−1 a (41) G(ζ) n ζn/k2−1 C′N N/κ1 A′N ζ N/k2−1. − Γ(n/k2) ≤ | | n=k0 X Notice that Σ′ ⋐ Σ(cf. Def. 2.1.2) and Σ ′ ) [ k π/(2κ ), +k π/(2κ )] ]0, 1/R [. − 2 1 2 1 × 0 1. In the variable Z, Estimate (39) reads

~~N−1 N an N/k1 A (42) F (Z) CN for all Z 1 − Zn/k2 ≤ Z N/k2 ∈ n=k0 | | X which, taking N = k0, implies that there exists constants M0,M1 > 0 such that~~

M1 (43) F (Z) M0, for all Z 1. | | ≤ Z k0/k2 ≤ ∈ | | In the integral of Formula (40) choose γ = ∂ and, for j = 1, 0, denote 1 ± ζU Gj(ζ) = 1/(2πi) F (U)e dU Z∂j so that G(ζ)= G−1(ζ)+ G0(ζ)+ G+1(ζ). The term G0(ζ) is a Riemann integral and determines a holomorphic function for all ζ. Due to Estimate (43) and the fact that k0/k2 > 1 the function F (U) is Lebesgue integrable on ∂ ∂ and, consequently, the functions G (ζ) are −1∪ +1 ±1 defined and holomorphic on the half-planes (ζ e±iω1 ) < 0 respectively. Thus, ℜ 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 145 the function G(ζ) is defined and holomorphic on the sector Σ, intersection of these two half-planes. 2. Denote G (ζ) = 1/(2πi) F (U)eζU dU so that G(ζ)= G (ζ)+ G (ζ) ± ℓ± − + for all ζ Σ. Parameterizing the paths ℓ by U = α + ue±iω1 we deduce from ∈ R ± Estimate (43) that G±(ζ) satisfies eαℜ(ζ) +∞ 1 G (ζ) M du for all ζ Σ. ± 1 k /k | | ≤ 2π 0 α + ue±iω1 0 2 ∈ Z | | Thus, there exists a constant c > 0 such that G(ζ) c eα|ζ| for all ζ Σ, | | ≤ ∈ and this proves the exponential growth of order 1 of G(ζ) at infinity on Σ. Prove that the Laplace transform (G)(Z) in direction θ = 0 is equal to L 0 F (Z) on the half-plane Z ; (Z) >R . By definition, (G)(Z) reads { ℜ 0} L 1 +∞ (G)(Z)= F (U)eζU dU e−ζZ dζ L 2πi 0 ∂ Z Z 1 ζ(U−Z) 1 + and the function F (U)e is in L (∂ 1 R ) when (Z) >R0. Indeed, ±iω1 × ℜ iθ parameterizing ∂±1 by U =(R0 + V )e and ∂0 by U = R0 e provides the estimates M eζ(R0 cos ω1−ℜ(Z))/ R + V k0/k2 for (U, ζ) ∂ R+ F (U)eζ(U−Z) 1 | 0 | ∈ ± × ζ(R0−ℜ(Z)) + ≤ ( M0 e for (U, ζ) ∂0 R . ∈ ×

By Fubini’s Theorem we can then write 1 +∞ (G)(Z) = F (U) eζ(U−Z)dζ dU L 2πi ∂ 1 0 1 Z F (U)Z = dU (∂ turns negatively around Z) 2πi −U Z 1 Z∂ 1 − = F (Z) (by Cauchy’s formula). 3. Use now the path λ with a radius R to be made explicit later. We want to estimate the quantity N−1 an n/k2−1 G(ζ) ζ Q−1 + Q0 + Q+1 − Γ(n/k2) ≤ n=k0 X N−1 n/k2 ζU where Qj = 1/(2π) F (U) an/U e dU , j = 1, 0 for λj − k0 ± ′ all ζ Σ . R P ∈ To estimate Q±1 use inequality (42) to write C +∞ eℜ(ζU) Q N N/k1 AN d U ±1 N/k ≤ 2π R U 2 | | Z | | 146 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

′ where (ζU)= ζU cos(θ′ ω ) and θ′ = arg(ζ) ω′. For ζ = ζ eiθ in Σ ℜ | | ± 1 | | ≤ | | then θ′ + ω satisfy π/2 < θ′ ω < 3π/2. Since Σ′ is a proper subsector 1 | ± 1| of Σ there exists c′ > 0 such that cos(θ′ + ω ) c′ for all ζ Σ′ and 1 ≤ − ∈ therefore, (ζU) c′ ζU for all ζ Σ′ and U λ . Using this estimate ℜ ≤− | | ∈ ∈ ±1 and the change of variable V = c′ ζ U in the latter integral we obtain | || | C +∞ e−V Q N N/k1 AN (c′ ζ )N/k2−1 dV ±1 N/k ≤ 2π | | ′ V 2 Zc |ζ|R C 1 +∞ N N/k1 AN (c′ ζ )N/k2−1 e−V dV ′ N/k ≤ 2π | | (c ζ R) 2 0 C 1 | | Z = N N/k1 AN (c′ ζ )−1 2π | | RN/k2 · ′ ′ For each ζ Σ choose R = N/ ζ (then, R>R0) and denote by C1 the ′ ∈ ′ | | ′ ′ constant C1 = C/(2πc ). It follows that Σ , Q±1 satisfies on Σ the estimate Q C′ N N/κ1 AN ζ N/k2−1 (recall 1/κ = 1/k 1/k ). ±1 ≤ 1 | | 1 1 − 2 To estimate Q parameterize λ by U = Reiθ with R = N/ ζ to obtain 0 0 | | ω1 iθ C N/k1 N 1 ℜ(N e ) Q0 N A e dθ N/k2−1 ≤ 2π R −ω1 C Z N N/κ1 NAN ζ N/k2−1 eN 2ω . ≤ 2π | | 1 ′ N N ′ N ′ Choosing A2 > Ae (so that NA e < Cst.A2 ) and C2 = Cst.Cω1/π we obtain Q C′ N N/κ1 A′ N ζ N/k2−1 on Σ′. 0 ≤ 2 2 | | ′ ′ ′ ′ ′ By adding these estimates and choosing A = A2 and C = 2C1 + C2 it follows that Estimate (41) is satisfied for all N 1 and all ζ Σ′. ≥ ∈ Suppose now that 1/k 2/k (hence, k π/(2k ) π). 1 ≥ 2 2 1 ≥ We observe that when ω1 passes the value π the expression of ω is changed from ω = π/2+ ω = k π/(2κ )+ ε to ω = 3π/2 ω = π k π/(2κ )+ ε . − 1 2 1 − 1 − 2 1 Hence, as ω increases through the value π (also k /k and k /κ increase) 1 2 1 2 1 the value of ω first increases up to π/2 (when ω1 = π) and then decreases. The sector Σ is no more large enough to prove the κ1-summability of g(ξ). We can pass through that difficulty by breaking 1 into finitely many subsectors of opening less than 2π. To this end, choose some directions θj and closed arcs J1,j of length less than 2π whose interiors make a covering of J1. From Cauchy’s Theorem the Borel transforms of F (Z) at the various directions θj are analytic continuations of each others and we can apply the previous proof 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 147

~~to each arc J1,j taking now the Borel transform in direction θj. This ends the proof of that part.~~

⊲ Prove that (ii) implies (i). Again we can restrict the study to the case when 1/k1 < 2/k2 (hence, k2π/(2k1) < π). We use the same notations as before. Assume Conditions 1, 2 and 3 are satisfied. Denote by a the type of ′ exponential growth of G(ζ) on Σ and, up to increasing the value of R0 (hence, up to shrinking the interval ]0, 1/R0[), suppose R0 >a. Choose a direction θ in Σ′ (i.e., θ ω′) and set 1 | 1| ≤ +∞eiθ1 −ζZ Fθ1 (Z)= G(ζ)e dζ. Z0 Condition 2 says that G(ζ) has exponential growth, say, of type a and it follows that the above definition of Fθ1 (Z) defines a holomorphic function Fθ1 (Z) on the half-plane (eiθ1 Z) > a bisected by θ at the distance a of 0. By ℜ − 1 Cauchy’s Theorem the functions F (Z) for the various values of θ Σ′ are θ1 1 ∈ analytic continuations from each other and we denote by F (Z) the function they define on the open sector 1 union of the half-planes associated with all θ Σ′. Observe that, since the opening of Σ′ is larger than k π/κ , the 1 ∈ 2 1 opening of 1 is larger than k2π/k1. By means of a rotation we can assume that θ1 = 0 and use Estimate (41) for ζ > 0. Given 0 < β < ε/2, denote by Πβ the sector Π = Z ; arg Z π/2 β and (Z) R >a . β { | | ≤ − ℜ ≥ 0 } ′ Notice that the condition on β implies that the sector 1 union of the Πβ’s ′ associated with the various directions θ1 Σ has opening more than k2π/k1 ′ ⋐ ∈ (and this is also the case for 1 1). Prove that Estimate (42): N−1 a AN F (Z) n CN N/k1 − Zn/k2 ≤ Z N/k2 n=k0 | | X (there exist A, C > 0) holds for all N and all Z in Πβ, for, the constants ′ involved are valid for any choice of θ1 in Σ .

+∞ n/k2−1 −ζZ n/k2 Since 0 ζ e dζ = Γ(n/k2)/Z we can write N−1 N−1 R a +∞ a F (Z) n = G(ζ) n ζn/k2−1 e−ζZ dζ n/k − Z 2 0 − Γ(n/k2) nX=k0 Z nX=k0 148 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY and then, N−1 an F (Z) P1 + P2 + P3 − Zn/k2 ≤ n=k0 X where

1/R0 N−1 an n/k2−1 −ζZ P1 = G(ζ) ζ e dζ,  0 − Γ(n/k2) | | Z n=k0  +∞ X  −ζZ  P2 = G (ζ)e dζ,   Z1/R0  N−1 + ∞ an n/k2−1 −ζZ  P3 = | | ζ e dζ.  Γ(n/k2) 1/R0  n=k0 Z  X From Estimate (41) we obtain, on Π ,  β +∞ ′ ′N N/κ1 N/k2−1 −ζℜ(Z) P1 C A N ζ e dζ ≤ 0 ′N Z ′ A N/κ1 = C N Γ(N/k2) (Z)N/k2 ℜ ′N ′ A N/κ1 C N Γ(N/k2) since (Z) Z cos β on Πβ ≤ ( Z cos(β))N/k2 ℜ ≥ | | | |N N A1 N/κ1 N/k2 A1 N/k1 C1 N N = C1 N for larger constants A1,C1 > 0. ≤ Z N/k2 Z N/k2 | | | | From Condition 2 we obtain, on Πβ, +∞ P c e(a−ℜ(Z))ζ dζ 2 ≤ Z1/R0 ce−(ℜ(Z)−a)/R0 ≤ ( (Z) a) a/Rℜ 0 − n −n n ce n e R0 n for all n> 0 and using (Z) Z cos β ≤ R1 a · Z cos α ℜ ≥ | | − N | | A2 N/k1 C2 N ≤ Z N/k2 | | 1/k by taking n = N/k ,A = R /(ek cos β) 2 and C = cea/R0 /(R a) and 2 2 0 2 2 1 − using N 1/k2 0 such that, for all n, a (44) n C′′nn/κ1 A′′n. Γ(n/k2) ≤

~~6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 149~~

It follows that P3 satisﬁes N−1 +∞ ′′ n/κ1 ′′n 1 N/k2−1 −ζℜ(Z) P3 C n A (R0ζ) e dζ since R0ζ 1 n/k2−1 ≤ 1/R0 R ≥ nX=k0 Z 0 ′′ ′′N ′′k0 (N−n)/k2 N/κ1 Γ(N/k2) C N max A ,A R0 N ≤ ( Z cos β)N/k2 N | | A3 N/k1 C3 N as before with large enough constants A3 and C3. ≤ Z N/k2 | | Adding these three estimates we obtain

N−1 N an A N/k1 (45) F (Z) C N on Πα − Zn/k2 ≤ Z N/k2 n=k0 | | X by setting A = max(A1,A2,A3) and C = C1 + C2 + C3. The constants A and C are independent of θ Σ′. Henceforth, estimate (45) is valid for all Z ′ 1 ∈ ∈ 1 and this proves the k1-summability of f(x) in direction θ0 since the opening ′ of 1 is larger than k2π/k1. This achieves the proof of the theorem.

The Tauberian Theorems of J. Martinet and J.-P. Ramis [MR89, Prop. 4.3](2) are easy corollaries of this theorem. Corollary 6.3.12 (Martinet-Ramis Tauberian Theorem 1) Let 0 < k1 < k2 and let I1 I2 be, respectively, a k1-wide and a k2-wide 1 ⊇ arc of S . Set s2 = 1/k2. If a series f(x) is both s2-Gevrey and k1-summable on I1 then it is k2- summable on I2 and the two sums agree on I2. e Observe that the assertion is not trivial since, according to Deﬁnition 6.1.6, being k2-summable on I2, compared to being k1-summable on I1, is a stronger condition to be satisﬁed on the smaller arc I2.

Proof. — It is suﬃcient to prove the theorem when I1 and I2 are closed of length π/k1 and π/k2 respectively. Let θ1 and θ2 be the bisecting directions of I and I ; they satisfy θ θ π/k π/k = π/k.A k -sum f (x) 1 2 | 1 − 2| ≤ 1 − 2 1 1 of f(x) exists on a larger open arc I1,ε containing I1 and lives in a sector 1,ε based on I1,ε. By Theorem 6.3.11 the k2-Borel transform g(ξ) of f1(x) in directione θ1 lives on an unbounded sector σ of opening π/k + ε bisected by θ1 and has there exponential growth of order k2. Moreover, g(ξ) is the unique function s-Gevrey asymptotic to g(ξ) on σ since the opening of σ is

~~(2) Caution: the notation s in that article correspondsb to our κ1. 150 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

larger than sπ = π/k. On another hand, since f(x) is s2-Gevrey, the formal Borel transform g(ξ) of f(x) is convergent. Its sum in the usual sense and the unique s-Gevrey asymptotic function g(ξ) muste necessarily agree. Denote by σc the union of theb sectore σ with the disc of convergence of g(ξ) and keep the notation g(ξ) for the function g(ξ) continued to σc. The domain σc contains the direction θ2. Using Deﬁnition 6.3.6 we can then concludeb that f(x) is k2-summable in direction θ2. In addition, from Theorem 6.3.11 we know that f1(x) is the analytice continuation of the k2-Laplace transform of g(ξ) in direction θ1. On another hand, it follows from Cauchy’s Theorem that the Laplace transforms of g(ξ) in directions θ1 and θ2 are analytic continuations from each other. Hence, the k2-sum f2(x) of f(x) coincides with f1(x) on I2 and the proof is achieved.

~~Corollary 6.3.13e (Martinet-Ramis Tauberian Theorem 2)~~

~~Let 0~~

~~k1 =1,k2 =2,I1 = I2 = θ ; θ < 3π/4 ande that, moreover, the 1-sum and the 2-sum { | | } when they both exist, agree. 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 151~~

Show that L(x) is not 1-summable in any direction θ satisfying π/4 θ ≤ ≤ 3π/4 mod π. Indeed, suppose π/4 <θ< 3π/4 mod π. The 1-Borel transform 1 L(ξ) B of L(x) satisﬁes ane equation obtained from the Leroy equation by substituting ξ (multiplication by ξ) for x2d/dx and d/dξ (derivation w.r.t ξ) for 1/x and so, it satisﬁese the equatione dY ξY +2 =2. dξ

After noticing that 1 L(0) = 0 we observe that 1 L(ξ) is the Taylor series of the entire B 2 B function Φ(ξ) = exp( ξ2/4) ξ et /4dt solution of the same equation. The function Φ(ξ) − 0 has exponential growthe of order exactly 2 in directione θ (since π/4 <θ< 3π/4). Hence, it R cannot be applied a Laplace transform and the series L(x) is not 1-summable in direction θ, and therefore, not 1-summable in all directions π/4 θ 3π/4 mod π. This property ≤ ≤ is coherent with the Tauberian Theorem 6.3.13 since,e as the series L(x) is divergent, it cannot be both 1- and 2- summable in almost all directions. e 6.3.4. Borel-Laplace summability and summable-resurgence. — We saw in Theorem 6.2.5 that solutions of linear differential equations with a unique level k are k-summable in any non anti-Stokes direction. In this section, we investigate deeper properties of such solutions called resurgence and summable-resugence in the case when k = 1. These notions of resurgence and summable-resugence are precisely defined and developed in [Sau, this volume] and in [Sau05] in the case of a one-dimensional lattice of singular points for the Borel transform; see also [CNP93]. They were introduced by J. Ecalle´ [E81,´ E85´ ] in a very general setting. They apply to a wide class of series, among which solutions of non linear differential equations or of difference equations. For a different approach in the linear differential case, hence in the case of a finite arbitrary set of singular points for the Borel transform, we refer to [LRR11]. The aim of this section is to show that the solutions of any linear differential equations with the unique level one are summable-resurgent.

Let D be a linear diﬀerential operator with meromorphic (convergent) 2 d coeﬃcients and let us expand it with respect to the derivation δ = x dx : D = b (x)δn + b (x)δn−1 + + b (x). n n−1 ··· 0 We suppose that its Newton polygon (D) at 0 has slopes 0 and 1 and N0 that f(x) is a series solution of the equation Dy = 0. The other formal λ λ q (1/x) solutions are either log-series fj(x)x j or log-exp-series fj(x)x j e j where fj(x) e C[[x]][ln x], λj C and qj(1/x)= aj/x, aj = 0. ∈ ∈ e − 6 e e 152 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

Deﬁnition 6.3.15.— The coeﬃcients aj of the (non-zero) determining polynomials qj(1/x) of the equation Dy = 0 are called Stokes values of Dy = 0 associated with f(x). These Stokes values indicate the anti-Stokes directions associated with e λ p f(x) as well as with any other log-series solution fj(x) x j ln (x). From Theorem 6.2.5 we know that f(x) is 1-summable in all directions but, possibly,e anti-Stokes directions. From the Borel-Laplacee viewpoint this means that, given a non anti-Stokes directionedθ, its associated Borel series f(ξ) is convergent and can be continued to a sector neighboring dθ with exponential growth of order 1. b We prove below that such properties can be extended to a much larger domain.

⊲ Suppose first that the coefficients bj(x) of D are algebraic (i.e., polynomials in x and 1/x). If the coefficients bj contain polynomial terms in x of maximal degree N > 0 we replace D by the operator D′ = x−N D. The operator D′ reads D′ = B (1/x)δn + B (1/x)δn−1 + + B (1/x) n n−1 ··· 0 −N with coefficients Bj(1/x)= x bj(x) that are polynomials in 1/x. Denote by v v their maximal degree in 1/x and set Bj(1/x) = (1/x )(γj + o(x)). By Borel transform, D′ is changed into the linear differential operator (cf. Sect. 4.3.2.2) d d d ∆′ = B ξn + B ξn−1 + + B n dξ n−1 dξ ··· 0 dξ and f(ξ) satisfies the equation ∆′y = 0. Lemma 6.3.16.— The set of the singular points of the equation ∆′y = 0 b S is the set of the Stokes values associatedb with f in the equation Dy = 0. Proof. — Since the Newton polygon at 0 of D (and D′) has the two slopesb 0 b and 1 we must have b0 = 0 and bn =0(cf. proof of Prop. 4.3.22). The operator ′ dk ℓ 6 ℓ dk ∆ has order v and, since dξk ξ = ξ dξk + “lower order terms” and b0 = 0, it reads n dv ∆′ = γ ξj + “lower order terms”. j dξv Xj=1 Hence, the singular points of the equation ∆′y = 0 are the zeroes of the n j polynomial j=1 γjξ = 0 which obviously vanishes for ξ = 0. However, this polynomial is also, up to a power of ξ, the 1-characteristicb polynomial of P D and we saw (cf. Sect. 4.3.2.3) that the non-zero Stokes values associated 6.3. THIRD APPROACH: BOREL-LAPLACE SUMMATION 153 with f(x) are the roots of the various characteristic polynomials. In our case, since the Newton polygon of D has no other slopes than 0 and 1, there is no othere characteristic polynomial than the 1-characteristic polynomial and we can conclude that the singular points of the equation ∆′y = 0 are all the Stokes values of the equation Dy = 0 including 0. It follows from the Cauchy-Lipschitz Theorem that f(ξ)b can be analytically continued along any path drawn in C which avoids the (finite) set of S singular points of the equation ∆′y = 0. The domain tob which extend the convergent series f(ξ) is then the Riemann surface, named , which is made RS of (the terminal end of) all homotopyb classes in C of paths issuing from \S 0 and bypassing allb points of . Only homotopically trivial paths are allowed S to turn back to 0. The surface looks very much like the universal cover of RS C but the fact that 0 is not a branch point in the first sheet (we always \S start with a convergent power series f(ξ)).

~~This property is named resurgenceb and we can state:~~

Lemma 6.3.17.— The series f(x) is resurgent with singular support the set of Stokes values associated with f(x) in the equation Dy = 0, i.e., its S Borel transform f(ξ) is convergente and can be analytically continued to the surface S . e R b Now, consider a sector on with image I ]R, + [ in C where the RS RS × ∞ arc I = θ <θ<θ is a bounded arc of directions and suppose it contains { 1 2} no singular point of the equation (I ]R, + [) = . One reaches × ∞ ∩ S ∅ RS from 0 following a 1-path γ of ﬁnite length in C . C \S We keep denoting by f(ξ) the analytic continuation of f(ξ) to . RS

~~Lemma 6.3.18.— The seriesb f(x) is summable-resurgentb on S , i.e., its R Borel transform f(ξ) has exponential growth of order 1 at inﬁnity on any e sector RS with bounded opening on S . b R~~

Proof. — This is a direct consequence of Proposition 4.3.22. If the opening of were not bounded we could turn inﬁnitely many times around adding RS ∞ exponential terms at each turn. Hence, the necessity to bound the opening of

~~RS . 154 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY~~

~~⊲ Now, turn back to the general case when D has meromorphic, not necessarily algebraic, coeﬃcients.~~

Theorem 6.3.19.— Let D be a linear differential operator with meromorphic (convergent) coefficients at 0 and suppose D has the unique level k = 1. Then, any series f(x) solution of Dy = 0 is summable-resurgent with singular support the Stokes values associated with f(x) in the equation Dy = 0 (cf. Def. 6.3.15). e e Proof. — The Algebraisation Theorem of Birkhoff (see [Bir09] or [Sib90, thm. 3.3.1]) says that to any linear differential operator D with meromorphic coefficients at 0 there exists a meromorphic transformation which changes D into an operator D′ with polynomial coefficients. The two equations Dy = 0 and D′y = 0 have the same determining polynomials, hence the same set S of Stokes values associated with a series f(x) in Dy = 0 or associated with its image after meromorphic transformation in D′y = 0. The operator D′ is relevant of Lemma 6.3.18. The Borel transforme of a convergent series is an entire function with exponential growth of order 1 at infinity. We thus have to prove that, if a function ϕ(ξ) is defined on all of with RS exponential growth of order 1 at infinity and g(ξ) is an entire function with exponential growth of order 1 at infinity then g ϕ is well defined on all of ∗ RS and has exponential growth of order 1 at infinity. Since g and ϕ are both analytic near the origin 0 their convolution product is well defined near 0 by the integral g ϕ(ξ)= ξ g(ξ t)ϕ(t)dy. Given any ∗ 0 − path γ from 0 to ξ in C (but its starting point 0), the integral g(ξ \S R γ − t)ϕ(t)dy is well defined and determines the analytic continuation of g ϕ R ∗ along γ. Changing the path γ into a homotopic one does not affect the result according to Cauchy’s Theorem. Hence, g ϕ is well defined on all of . The ∗ RS fact that this function has exponential growth of order 1 at infinity follows from the fact that the convolution of exponentials eAξ and eBξ of order 1 is itself a combination of exponentials of order 1.

Remark 6.3.20. — The previous theorem is valid for all series appearing in a formal fundamental solution of a linear differential equation with meromorphic coefficients and unique level k = 1 even those that come with a complex power of x or in a formal-log sum. This follows from the fact that they are themselves solution of a (another) linear differential equation of the same type. This is easily seen on systems: in a formal fundamental solution 6.4. FOURTH APPROACH: WILD ANALYTIC CONTINUATION 155

~~F (x)xL eQ(1/x) the factor F (x) is solution of the homological system, itself with meromorphic coeﬃcients and unique level 1. e e~~

6.4. Fourth approach: wild analytic continuation The fourth deﬁnition of k-summability deals with wild analytic continuation, that is, continuation of the series in the inﬁnitesimal neighborhood of 0 (cf. Sect. 4.5).

6.4.1. k-summability. — A Gevrey series is a germ at 0 of the sheaf . F We call wild analytic continuation any of its continuations as sections of . F A series f(x) which is k-summable on a k-wide arc I can be wild analytically continued to a domain containing the disc D(0,k) and the sector (θ,k′); θ { ∈ I and 0 e

C[[x]] := lim C[[x]] = C[[x]] C[[x]] s+ ←− s+ε s+ε ⊃ s ε→0+ ε>0 \ and is thus, bigger than C[[x]] . As for the set of global sections of over the s F closed disc D(0,k), it is isomorphic to

C[[x]] := lim C[[x]] = C[[x]] C[[x]] s− −→ s−ε s−ε ⊂ s ε→0+ ε>0 [ and smaller than C[[x]]s (cf. Prop. 4.5.3). The right domain lies in between D(0,k) and D(0,k). It can be made explicit in the sheaf space (Xk, k) F since, indeed, the set of global sections of k over the closure D(0, k, 0 ) in F { } Xk of the open disc D(0, k, 0 ) is isomorphic to C[[x]] (cf. Prop. 4.5.5). { } s We can then state the following new deﬁnition of k-summability:

Deﬁnition 6.4.1 (k-summability).— Let I be a k-wide arc of S1 n (cf. Def. 6.1.2). A series f(x) = n≥0 an x is k-summable on I if it can be wild analytically continued to a domain containing the closed disc P D(0, k, 0 ) and the sector I e ]0, + ]. We call such a domain a k-sector in { } × ∞ Xk.

The deﬁnition above being the exact translation of Ramis-Sibuya definition of k-summability it is equivalent to all previous deﬁnitions of k- summability. 156 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

~~Figure 13. Domain for a k-sum in Xk (in white)~~

~~6.4.2. Applications. — 1. Tauberian Theorems — Let us revisit the Martinet-Ramis Tauberian Theorem 1 (Cor. 6.3.12) from the viewpoint of wild analytic continuation.~~

Consider a k1-summable series f(x) on I1. In the viewpoint of wild analytic continuation this property translates in the space (Xk1 , k1 ) F (cf. Sect. 4.5.2 p. 93) as the conditione that the series f(x) admits a continuation as a section of the sheaf k1 to the k -sector = F 1 k1,I1 D(0, k , 0 ) I ]0, + ] (see Fig. 13). The fact that it is ke-summable on { 1 } ∪ 1× ∞ 2 I has a similar interpretation in the space (Xk2 , k2 ). To interpret both we 2 F need to work in the space (Xk1,k2 , k1,k2 ). F The Tauberian Theorem says that, given k < k and I I , the fact 1 2 2 ⊂ 1 that f(x) be k1-summable on I1 (i.e., that it can be continued to the k1-sector

k1,I1 ) and that it be also s2-Gevrey (i.e., that it can be continued to the disc D =e D(0, k , 0 )) implies that it can be continued to the k -sector . k2 { 2 } 2 k2,I2 Clearly, is included in D . k2,I2 k1,I1 ∪ k2

~~k1,k2 Figure 14. k1,I1 (in yellow) and k2,I2 (hachured) in X 6.4. FOURTH APPROACH: WILD ANALYTIC CONTINUATION 157~~

The theorem asserts that, on the intersection D , the two contin- k1,I1 ∩ k2 uations agree. This is the case on Dk1 since there is a unique continuation of f(x) to Dk1 (cf. R-S Cor. 6.2.2 p. 119). The compatibility of the two continuations on D D means that their diﬀerence belongs to k2 \ k1 ∩ k1,I1 thee space H0 I , ≤−k1 / ≤−k2 . The Relative Watson’s Lemma below (Thm. 1 A A 8.2.1 p. 175) asserts that such a space reduces to the null section. Hence, the two continuations agree and deﬁne a k2-sum of f(x) on I2.

~~e 2. Functions of k-summable series~~

Proposition 6.4.2.— Let be given a k-wide arc I and r series f1(x),..., fr(x) that are k-summable on I with k-sums f1(x),...,fr(x) respectively. Assume that f1(0) = = fr(0) = 0. e e ··· r+1 If g(x,y1,...,yr) is an analytic function on a neighborhood of 0 in C then, the seriese g(x, f1(xe),..., fr(x)) is k-summable on I with k-sum g(x,f1(x),...,fr(x)). e e Proof. — According to Proposition 2.3.6 the expression g(x, f1(x),..., fr(x)) determines a well-defined s-Gevrey series, hence, a germ at 0 of the sheaf k F which can be continued to the closed disc D(0, k, 0 ). e e { } The series f1(x),..., fr(x) being k-summable on I and vanishing at x = 0 can be continued to the sector I ]0, + ] with values in an arbitrary small × ∞ neighborhood ofe 0. The functione g being holomorphic on a neighborhood of 0 the series g x, f (x),..., f (x) can also be continued to the sector I ]0, + ] 1 r × ∞ with analytic continuation g x,f (x),...,f (x) . 1 r e e 3. Summability of solutions of differential equations Let f(x) be a series solution of a linear differential equation (or system). The fences to the wild analytic continuation of f(x) in X or in any space k k1,ke2 X ,X ,... are the big points of the exponentials exp(qj(1/x)) j∈J appearing in a formal fundamental solution. Indeed,e when a direction passes a big point it exits the definition domain of the associated exponential and flat terms become undefined. Recall that, in X, the big points associated with an exponential of degree k are the closed arcs of length π/k bisected by the anti-Stokes directions of the exponential (directions of maximal decay) and lying on the circle of radius k in X. In Xk they are arches based on the previous arcs. Below, are drawn the big points of two exponentials, one of degree 1 and one of degree 2 in X. 158 CHAPTER 6. FOUR EQUIVALENT APPROACHES TO K-SUMMABILITY

A series solution of a linear diﬀerential equation where only these two exponentials appear is k-summable on any k-sector containing none of these big points. Also, it is (k1,k2)-summable on any (k1,k2)-sector containing none of these big points. In particular, one can check easily that it is (1, 2)- summable in almost all directions (here, all directions but the three anti-Stokes directions). When f(x) is a series solution of a non linear diﬀerential equation then the same procedure applies to the linearized equation along f(x). e e CHAPTER 7

~~TANGENT-TO-IDENTITY DIFFEOMORPHISMS AND BIRKHOFF NORMALISATION THEOREM~~

7.1. Introduction This chapter deals with the conjugacy of tangent-to-identity germs of diffeomorphisms at 0. It aims at showing another example (not solution of a differential equation) where the Gevrey cohomological analysis is also efficient. We consider the by-now classical case of a germ of “translation” x g : x g(x)= 7−→ 1+ x· As a homography, g is defined over the whole Riemann sphere C. In the chart of infinity, setting z = 1/x and G(z) = 1/g(x), the germ g reads G : z G(z)= z + 1 7−→ hence, the name of translation. Convention. — As previously, we denote by x the coordinate about 0 and by z = 1/x the coordinate about infinity. We denote by the same letter a given germ in the chart of 0 and in the chart at infinity, using a small letter at 0 and the corresponding capital one at infinity. In this context, the formal and meromorphic gauge transformations of the classification of linear differential systems are replaced by formal and conver- n gent tangent-to-identity diffeomorphisms h(x) = x + n≥2 cnx acting on g by conjugacy, that is, by changing g into h−1 g h. e ◦ ◦ P Definition 7.1.1.— A germ f is formally conjugated (or analytically e e n conjugated) to g if there exists h(x)= x + n≥2 cnx formal (or convergent) satisfying the conjugacy equation e P (46) h f = g h. ◦ ◦ e e 160 CHAPTER 7. TANGENT-TO-IDENTITY DIFFEOMORPHISMS

One can check that such an h exists if and only if f has the form f(x)= x x2 + x3 + a xn e− n nX≥4 and p h(x)= x + cpx Xp≥2 is unique after c2 is ﬁxed, say,e to c2 = 0. In the chart of inﬁnity, setting z = 1/x and F (z) = 1/f(x), the condition reads A F (z)= z +1+ n (observe A = 0) zn 1 nX≥2 and H is unique in the form

Cp e H(z)= z + zp · Xp≥1 Notice that, in its formal class,e g has the particularity of being best behaved with respect to iteration and, thus, plays the role of a normal form. p From now on, the diﬀeomorphisms h(x) = x + p≥2 cpx by which we conjugate are supposed to satisfy c = 0. We denote the group of the so 2 P normalized germs of formal tangent-to-identitye diﬀeomorphisms of C at 0, endowed with composition, by

~~G = x + c xn C[[x]] n ∈ n nX≥3 o e and the subgroup of convergent germs of G by~~

G = x + c xn C x . n e ∈ { } n nX≥3 o With this normalization, a conjugacy map h when one exists is unique. It might be divergent although f and g are both convergent. One can prove, for instance, that a sufficient condition for h toe be divergent is that f be an entire function. As for linear differential systems the analytice classification of the conjugacy classes of diffeomorphisms is performed inside each formal class with a given normal form, here g. Our aim is not to longly develop that classification but to give a proof of the main point in the given example of the translation g, that is to say, to prove that the conjugacy maps h of g are 1-summable series.

~~e 7.1. INTRODUCTION 161~~

A natural approach consists in analyzing the Borel transform of h(x) following so J. Ecalle´ [E74´ ] (cf. D. Sauzin, section 14, this volume). We choose to develop here a sectorial approach due to Kimura [Kim71, Thm.e 6.1] (see also [Bir39, première partie, 5]); the proof is based on the Ramis-Sibuya § Theorem (Thm. 6.2.1) after constructing an adequate 1-quasi-sum. We consider the following sheaves: ⊲ = f ; Tf G the subsheaf of made of (normalized) tangent- G ∈A ∈ A to-identity germs of diffeomorphisms (Recall that is the sheaf over S1 of A germs of asymptotic functionse at 0; cf. Sect. 3.1.5) and ⊲ <0 = f ; Tf = id the subsheaf of made of its flat germs (“flat” G ∈G G in a multiplicative context means “asymptotic to identity”). Equipped with composition law, is a sheaf of non commutative groups G and <0 a subsheaf of groups. G Given 0 < α < π we consider in the chart of infinity the sectors

~~∆ (α,R)= z ; α< arg(z R) <α , + − − ∆ (α,R)= z ; π α< arg(z + R) < π + α . − −~~

~~∆+(α,R) and ∆+(α,R) are symmetric to each other with respect z = 0.~~

~~We denote by δ+(α,R) and δ−(α,R) their image in the coordinate x = 1/z.~~

~~Figure 1~~

~~Given g a germ of diﬀeomorphism we denote its pth power of compsition by gp = g g g . ◦ ◦ · · · ◦ p times | {z } 162 CHAPTER 7. TANGENT-TO-IDENTITY DIFFEOMORPHISMS~~

7.2. Birkhoff-Kimura Sectorial Normalization Although we state the theorem in a chart of 0 (coordinate x), as we are use to do it, it is worth, taking into account the very simple expression of G(z), to perform the proof in the chart of infinity (coordinate z). Let us start with a technical lemma. Lemma 7.2.1.— Let 0 < α < π and R > 1 be given. For all m N∗, 0 0 ∈ there exists a constant c> 0 which depends on α0 and m but not on R0 such that, 1 c m for all z ∆+(α0,R0). z + p m+1 ≤ z ∈ Xp≥0 | | | | Proof. — The proof is elementary. We compare the sum to an integral as soon as possible, i.e., as soon as the general term of the series decreases and we estimate the extra terms. Given z ∆ (α ,R ) let us denote by p(z) + 1 0 the smallest integer ∈ + 0 0 ≥ such that (z + p(z) + 1) > 0. Notice that p(z) max 0, z cos(π α ) and ℜ ≤ | | − 0 z R sin α for all z in ∆ (α ,R ). We split the series into | | ≥ 0 0 + 0 0 p(z)+1 1 1 1 = + m+1 m+1 m+1 · z + p p=0 z + p z + p Xp≥0 | | X | | p≥Xp(z)+2 | | We claim first that z + p z sin α > 0 for all p N and z | | ≥ | | 0 ∈ ∈ ∆ (α ,R ); for, z + p z when (z) 0 and z + p (z)= z sin θ + 0 0 | |≥| | ℜ ≥ | |≥ℑ | | ≥ z sin α when (z) < 0 since then π/2 <θ<α . It follows that | | 0 ℜ 0 p(z)+1 1 1 p(z) + 1 c + 1 m+1 ≤ m+1 ( z sin α )m+1 ≤ z m p=0 z + p z 0 X | | | | | | | | for a constant c1 depending on α0 and m but not on R0 > 1. Indeed, we have 1 p(z) + 1 1 cos(π α ) 1 + + − 0 + z z (sin α )m+1 ≤ R sin α (sin α )m+1 R (sin α )m+2 | | | | 0 0 0 0 0 0 m+2 and, since R0 > 1, we can choose c1 = 3/(sin α0) . Starting from p = p(z) + 1 the function p 1 decreases and we 7→ |z+p|m+1 have 1 +∞ dp +∞ dq m+1 m+1 = m+1 z + p ≤ p(z)+1 z + p 0 z + p(z)+1+ q p≥Xp(z)+2 | | Z | | Z | | 1 +∞ dr c = 2 z + p(z) + 1 m (1 + r)m+1 ≤ z m | | Z0 | | 7.2. BIRKHOFF-KIMURA SECTORIAL NORMALIZATION 163 for a constant c = 1/(sin α )m; indeed, z + p(z) + 1 z sin α and 2 0 | | ≥ | | 0 since m 1, we can write ≥ +∞ dr +∞ dr = 1. (1 + r)m+1 ≤ (1 + r)2 Z0 Z0 Hence, the result if one chooses the constant c = c1 + c2.

Theorem 7.2.2 (Birkhoff-Kimura sectorial normalization) Let ϕ be a flat diffeomorphism over a proper sub-arc I+ =] α , +α [ of α0 − 0 0 S1 (i.e., 0 <α < π and ϕ H0(I+ ; <0)). 0 ∈ α0 G Then, the diffeomorphism g = ϕ g belongs to H0(I+ ; ) and is uniquely 1 ◦ α0 G conjugated to g via a section of <0: there exists a unique φ H0(I+ ; <0) G + ∈ α0 G such that φ g = g φ on I+ . + ◦ 1 ◦ + α0 Symmetrically, denote by I− =[ α +π,α +π] the arc opposite to I+ on S1 α0 − 0 0 α0 and suppose that ϕ H0(I− ; <0). Then, there exists a unique φ H0(I− ; <0) ∈ α0 G − ∈ α0 G such that φ g = g φ on I− . − ◦ 1 ◦ − α0 + − Proof. — We make the proof over Iα0 . The proof on Iα0 is similar when applied to g−1 and g−1. The fact that g = ϕ g be a diffeomorphism on I+ 1 1 ◦ α0 and have a Taylor expansion is clear since so do ϕ and g. Its Taylor expansion is equal to Tg = Tϕ Tg = id Tg = Tg. Turn now to the variable z and 1 ◦ ◦ denote by the corresponding capital letters the diffeomorphisms in the chart of infinity. Given α<α choose α ]α,α [ and R > 1 so that G (z) be well 0 1 ∈ 0 1 1 defined on ∆ (α ,R ). Denoting K(z)= G (z) G(z) and φ (z)= z +ψ (z) + 1 1 1 − + + the condition φ g = g φ becomes K(z)+ ψ G (z) ψ (z) = 0. A + ◦ 1 ◦ + + ◦ 1 − + solution will be given by ψ (z)= K Gp(z) if we prove that the series + p≥0 ◦ 1 K Gp(z) converges to a holomorphic function asymptotic to 0 at infinity. p≥0 ◦ 1 P P ⊲ Given R > R1 + 2, the function K(z) being asymptotic to 0 on ∆ (α ,R ) it satisfies: for all m N, there exists a> 0 such that + 1 1 ∈ a (47) K(z) on ∆+(α,R1 + 1) ∆+(α,R 1). ≤ z m+1 ⊃ − | | The constant a depends on m, α1 and R1 but not on R. Below, R>R1 + 2 will be chosen conveniently large. 164 CHAPTER 7. TANGENT-TO-IDENTITY DIFFEOMORPHISMS

~~⊲ Prove that, there exists a constant A a and independent of R such ≥ that,~~

~~′ A (48) sup K(z ) m+1 |z−z′|≤sin α ≤ z · z∈∆+(α,R) | |~~

~~Indeed, the conditions z ∆ (α,R) and z z′ sin α imply z′ ∈ + | − | ≤ ∈~~

~~Figure 2~~

∆ (α,R 1). We can then apply Condition (47) to yield + − a a A K(z′) ≤ z′ m+1 ≤ z sin α m+1 ≤ z m+1 | | | |− | | m+1 with A = a R 1 since R1−1 z z R1 sup | | sup | | = z sin α ≤ z sin α R 1· z∈∆+(α,R) | |− |z|≥R1 sin α | |− 1 −

⊲ Choose R so large that, for all m 1, ≥ 1 (49) A sin α for all z ∆+(α,R). z + p m+1 ≤ ∈ Xp≥0 | | Such a choice is possible. Indeed, from Lemma 7.2.1 applied to ∆+(α1,R1), we obtain 1 Ac A m z + p m+1 ≤ z Xp≥0 | | | | Ac and the constant Ac does not depend on R. Now, maxz∈∆+(α,R) |z|m = Ac Ac (R sin α)m . Assuming R large enough so that R sin α > 1 then, (R sin α)m Ac ≤ R sin α which is independent of m and can be made arbitrarily small by choosing R large. 7.2. BIRKHOFF-KIMURA SECTORIAL NORMALIZATION 165

~~⊲ Prove by induction on p that, for all p 1 and all z ∆ (α,R), ≥ ∈ + p−1 1 (50) Gp(z) (z + p) A 1 − ≤ m+1 q=0 z + q X | |~~

(recall the notation gp = g g g times). ◦ ◦ · · · ◦ p When p = 1, the inequality reads K(z) A/ z m+1 and follows from | {z } ≤ | | Condition (47) with a < A. Suppose Condition (50) valid up to p. Then, from Condition (49), we get Gp(z) (z + p) sin α. And this implies that: | 1 − | ≤ p (i) lim G (z)= for all z ∆+(α,R); p→∞ 1 ∞ ∈ (ii) Gp(z) ∆ (α,R) since z + p ∆ (α,R + p) ∆ (α,R + 1); 1 ∈ + ∈ + ⊂ + A (iii) K Gp(z) (Estimate (48) applied to z′ = Gp(z) and ◦ 1 ≤ z + p m+1 1 z + p for z). | |

Since Gp(z) ∆ (α,R) it can be applied G = G+K. We can then write 1 ∈ + 1 Gp+1(z)= G Gp(z) + K Gp(z) = Gp(z)+1+ K Gp(z) 1 1 1 1 ◦ 1 from which we deduceGp+1(z) (z + p + 1) = Gp(z) (z + p)+ K Gp(z). 1 − 1 − ◦ 1 Applying the recurrence hypothesis and Condition (iii) at rank p we obtain

~~p−1 1 A Gp+1(z) (z + p + 1) A + 1 − ≤ m+1 m+1 q=0 z + q z + p X | | | | which is Condition (50) at rank p + 1.~~

⊲ Conclude on ψ . Condition (iii) for all p and m 1 proves that the + ≥ series K Gp(z) ◦ 1 Xp≥0 converges uniformly on compact sets of ∆ (α,R). The functions K Gp being + ◦ 1 holomorphic, the sum Ψ (z)= K Gp(z) is holomorphic on ∆ (α,R). + p≥0 ◦ 1 + Moreover, for all m N∗ and all z ∆ (α,R) (Recall that α and R do not ∈ P ∈ + 166 CHAPTER 7. TANGENT-TO-IDENTITY DIFFEOMORPHISMS depend on m) , there exist constants A and c> 0 such that A Ψ+(z) (Condition (iii)) ≤ z + p m+1 p≥0 | | X Ac (Lemma 7.2.1 for ∆ (α,R)) ≤ z m + | | which shows that ψ+(z) is asymptotic to 0 at inﬁnity.

⊲ Proving the uniqueness of the solution resumes to proving that the equation ψ G1 ψ = 0 has a unique solution asymptotic to 0 on ∆+(α,R). ◦ − p And indeed, if we iterate the equation we obtain ψ G1(z) ψ(z) = 0; letting ◦ p − ′ p tend to inﬁnity, we obtain ψ(z) = lim ψ G (z) = lim ′ ψ(z ) p→+∞ ◦ 1 z →∞ according to Condition (i). Hence, ψ(z) = 0 for all z ∆ (α,R) and the proof is achieved. ∈ + Actually, Birkhoﬀ [Bir39] and Kimura [Kim71] stated the theorem in the following form (see also [Mal82] and [E74´ ]).

Corollary 7.2.3 (Birkhoff-Kimura).— Consider the conjugacy equation (46) h f = g h. ◦ ◦ With notations as before, there exist unique diffeomorphisms h H0(I+ ; ) e e + ∈ α0 G and h H0(I− ; ) such that − ∈ α0 G h f = g h and T h = h on I+ , + ◦ ◦ + 0 + α0 − h− f = g h− and T0 h− = h on Iα . ◦ ◦ e 0 where T h stands for “Taylor expansion of h at 0”. 0 e + Proof. — Again, we develop the proof over Iα0 . We denote by δ+(α,R) the image in the chart of 0 of the domain ∆+(α,R) as built in the proof of Theorem 7.2.2. The Borel-Ritt Theorem (Thm 2.4.1 (i)) provides a function h holomorphic on δ+(α,R) and with Taylor expansion Th(x)= f(x) at 0. Consider the function f = h f h−1. It has an asymptotic expansion on δ (α,R) given 1 ◦ ◦ + by h f h−1 = g according to the conjugacy equatione (46). Hence, there ◦ ◦ exists ϕ = f g−1 which is flat and satisfies f = ϕ g. 1 ◦ 1 ◦ eBirkhoff-Kimurae Theorem 7.2.2 applied to f1 and g provides a ψ+ asymptotic to 0 and satisfying ψ f = g ψ and h = ψ h solves the problem + ◦ 1 ◦ + + + ◦ on δ+(α,R). Uniqueness is proved similarly as for ψ+ and is valid on δ(α,R) 7.3. INVARIANCE EQUATION OF g 167

for all α<α0 (whereas R might depend on α) . Hence the existence and uniqueness of h as a section of the sheaf over I+ . + G α0 − Symmetrically, we prove the existence and uniqueness of h− over Iα0 by the same method.

~~When α0 > π/2 the domains of deﬁnition of h+ and h− overlap across the two imaginary directions.~~

~~Figure 3~~

7.3. Invariance equation of g The invariance equation (51) u g = g u ◦ ◦ of g is a particular case of the conjugacy equation (46). Hence, it admits the unique solution u =Id in G and, given 0 < α0 < π, it admits a unique solution u section of the sheaf over I+ =] α , +α [ and a unique solution + G α0 − 0 0 u over I− =]π α , π + α [e asymptotic to g. But due to their uniqueness, − α0 − 0 0 since Id is a solution everywhere, then u+ and u− are both equal to Id. The situation is diﬀerent in a neighborhood of the imaginary axis where there might exist non trivial germs of solutions. And indeed, the solution h−1 h might − ◦ + be non trivial depending on f. In this section we study the behavior of germs of ﬂat solutions near the two imaginary half axis. Proposition 7.3.1.— Let = x < r ; β < arg x < π β with 1 | | 1 − 0 < β < π/2 be a sector with vertex 0 neighboring the positive imaginary axis. Any solution u ( ) of the invariance equation (51) is exponentially ∈G 1 168 CHAPTER 7. TANGENT-TO-IDENTITY DIFFEOMORPHISMS

ﬂat of order 1 on 1. The same result holds on a sector = x R2 = ; β < arg z < π β , | | r2 − the solution u is changedn into U and we set U = Id+V . Witho these notations the invariance equation reads (52) V (z + 1) = V (z) whose solutions are the 1-periodic functions. Hence, to solutions U ( ) ∈ G 2 there correspond functions V of the form V (z) = ν(e2πiz) that satisfy lim V (z) = 0. z→∞

~~Figure 4~~

Consider, in , a vertical half-stripe [iR, 1+iR[ ]iR, +i [ with width 1 2 × ∞ (see Fig. 4, p. 168). It’s easily checked that its image by the map z t = e2πiz 7→ is a punctured disc Ω2 centered at 0 in C. Moreover, a fundamental system of neighborhoods of infinity in 2 is sent on a fundamental system of neighborhoods of 0. Hence, the condition lim ν(e2πiz) = 0 is equivalent to z→∞ z∈Σ1 limt→0 ν(t) = 0. Consequently, by the Inexisting Singularity Theorem, ν can be continued into a holomorphic function at 0. Now, suppose ν is not identically 0. Then, it has a finite order, say k at 0 (denote ν(t)= O(tk)). This implies that V (z)= O(e−2πkℑ(z)) as z tends 7.4. 1-SUMMABILITY OF THE CONJUGACY SERIES H 169 e to infinity in . However, on , one has (z) > z sin β and consequently, 2 2 ℑ | | V (z)= O(e−2πk sin(β)|z|). Thus, V has (uniform) exponential decay of order one on 2 at infinity and so does the solution u(x) = 1/ 1/x+V (1/x) at x = 0 on . 2

7.4. 1-summability of the conjugacy series h Recall that the conjugacy equation e h f = g h (46) ◦ ◦ admits a unique formal solution h(x) in G. The 1-summability of h is now straightforward. e e e Theorem 7.4.1.— The series h(x) is 1-summable with singular directions the two imaginary half-axis. e Proof. — Let π/2 < α < π and consider the solutions h (x) H0(I− , ) 0 + ∈ α0 G and h (x) H0(I+ , ) of Equation (46), asymptotic to h(x)(cf. Cor. 7.2.3). − ∈ α0 G The non Abelian 1-cocycle defined by h = h−1 h on ]π α ,α [ and 1 − ◦ + − 0 0 by h = h−1 h on ] α ,α π[ satisfies the invariancee equation (51). 2 + ◦ − − 0 0 − Denote h1 = Id+u1 and h2 = Id+u2. It follows from the previous section that u and u are exponentially flat of order one on ]π α ,α [ and ] α ,α π[ 1 2 − 0 0 − 0 0 − respectively. To apply the Ramis-Sibuya Theorem to h(x) on the covering =(I+ ,I− ) I α0 α0 of S1 we must prove that the Abelian 1-cocycle equal to h h on ]π α ,α [ + − − − 0 0 and h h on ] α ,α π[ is exponentiallye flat. And indeed, from the − − + − 0 0 − form h = h−1 h = Id+u and h = h−1 h = Id+u of h and h we 1 − ◦ + 1 2 + ◦ − 2 1 2 deduce that h h = h u on ]π α ,α [, + − − − ◦ 1 − 0 0 ( h h = h u on ] α ,α π[. − − + + ◦ 2 − 0 0 − Hence, the Abelian 1-cocycle is exponentially flat of order one since so are u1 and u2 while h− and h+ are asymptotic to the identity. Since α0 can be chosen arbitrarily close to π we can conclude that the series h(x) is 1-summable in all direction but the two imaginary half axis. One proves that these cocycles are not trivial in general while h(x) is divergent;e the non Abelian 1-cocycle (h1,h2) classifies the analytic classes of diffeomorphisms f(x) formally conjugated to g(x). e

~~CHAPTER 8~~

~~SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY~~

8.1. Introduction and the Ramis-Sibuya series We may observe that the examples of series given in the previous chapters that are solution of linear differential equations are all k-summable for a convenient value of k. In Theorem 6.2.5 sufficient conditions are stated for the k-summability of solutions of linear differential equations (k-summability must be understood there in its global meaning, that is, k-summablility in almost all directions). Recall that Corollary 6.3.13 asserts that a series both k1- and k -summable for two distinct values k = k of k is necessarily convergent. 2 1 6 2 Though, such a result is no longer valid if one considers k1- and k2-summability in a given direction θ: as shown in Example 6.3.14, the Leroy series L(x) is both 1- and 2-summable in all directions θ ] π/4, +π/4[ mod π. ∈ − A first natural question is to determine whether any series solutione of a linear differential equation is k-summable for a convenient value of k. This question, known under the name of Turrittin problem although Turrittin after Trjitzinsky, Horn and al. formulated the question in different terms, received a negative answer by J.-P. Ramis and Y. Sibuya in 1984 (published later [RS89]) through a counter-example (cf. Exa. 8.1.1). A more intricate summation process called multisummation had become necessary. The counter-example given by J.-P. Ramis and Y. Sibuya with a proof of the fact that the Ramis-Sibuya series is k-summable for no k > 0 is as follows.

~~Example 8.1.1 (Ramis-Sibuya series).— The Ramis-Sibuya series is the series RS(x)= E(x)+ L(x)~~

~~f e e 172 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY~~

sum of the Euler series E(x) = ( 1)n n! xn+1 and of the Leroy series n≥0 − L(x)= ( 1)n n! x2n+2 deduced from the Euler series by substituting x2 for x n≥0 − P (cf. Exa. 2.2.4 and 6.3.14). Frome the Euler equation x2y′ + y = x and the Leroy equa- P tione x3y′ +2y =2x2 one deduces that the Ramis-Sibuya series satisﬁes the Ramis-Sibuya equation RS(y)=4x +2x2 + 10x3 3x4 − where the operator RS reads RS = x5(2 x)d2/dx2 + x2(4+5x2 2x3)d/dx + 2(2 x + x2). − − − It is worth to notice that the operator RS admits the following Newton polygon with the two slopes 1 and 2. We will see that this indicates that the series solution of the Ramis-Sibuya equation are all, at worst, (1,2)-summable as deﬁned in the next sections.

~~Figure 1. Newton polygon of the Ramis-Sibuya operator~~

Check that the series RS(x) is k-summable for no k > 0. Indeed, as we saw earlier (cf. Com. 6.1.9 and Sect. 2.2.2), the Euler series E(x) is 1-summable in all direction but the direction θ = π. As a consequence,f the Leroy series L(x) is 2-summable in all direction but the directions θ = π/2. We saw in Examplee 6.3.14 that the Leroy series, and then ± also the Ramis-Sibuya series, is 1-summable in the directionse θ ] π/4, +π/4[ mod π and ∈ − in these directions only. In particular, the Ramis-Sibuya series is not 1-summable. On another hand, the Euler series E(x), and then also the Ramis-Sibuya series, is 2-summable in no direction since its 2-Borel transform does not converge. In particular, the Ramis- Sibuya series is not 2-summable.e Consider now a direction θ ]π/4 , 3π/4[ mod π and show ∈ that RS(x) is k-summable for no other value of k > 0 in direction θ. This is the case for k > 1 (and in any direction) from the same argument as for k = 2: the k-Borel transform of RS(fx) does not converge. Suppose there exists k < 1 such that RS(x) be k-summable in direction θ. Then, since RS(x) is a 1-Gevrey series, the Tauberian Theorem (Thm. 6.3.12)

off Martinet-Ramis [MR89] (taking k1 = k < 1 and k2 = 1) wouldf imply that RS(x) be 1-summable in directionf θ. Hence, the contradiction and we can conclude that RS(x) is k-summable for no k > 0 since this is the case in all direction θ ]π/4 , 3π/4[ modfπ. ∈ Let us also sketch another proof of this latter fact which relies on the study off the Stokes phenomenon for RS(x) and makes no use of the Tauberian Theorem. A k-sum of RS(x) would be a solution of the Ramis-Sibuya equation (cf. Prop. 6.1.10). The space of solutions of the Ramis-Sibuyaf equation is the sum of the spaces of solutions of the Euler f 8.1. INTRODUCTION AND THE RAMIS-SIBUYA SERIES 173

equation and of the Leroy equation. The Stokes arcs are the arc [π/2 , 3π/2] inherited from E(x) and the two arcs [π/4 , 3π/4] mod π inherited from L(x). An arc centered at the chosen direction θ ]π/4 , 3π/4[ mod π with length > π/k > π contains a Stokes arc ∈ of L(xe) and then all solutions of the Leroy equation asymptotice to L(x) must exhibit a Stokes phenomenon with non trivial Stokes automorphisms on that arc. Moreover, the discontinuities,e i.e., the Stokes automorphisms of the Leroy series beinge given in terms of exponentials of order 2 cannot be compensated by the Stokes automorphisms of the Euler series that are given in terms of an exponential of order 1. Consequently, there exists no solution of the Ramis-Sibuya equation asymptotic (and especially, k-asymptotic) to RS(x) on such an arc and we can again conclude that RS(x) is not k-summable in any direction θ ]π/4 , 3π/4[ mod π. f ∈ f A natural candidate for the sum of RS(x) is the asymptotic function E(x)+L(x) obtained by adding the 1-sum of E(x) to the 2-sum of L(x). But that choice is not so trivial as we will see soon. The (1,2)-summabilityf of RS(x) following J. Ecalle’s´ approach is widely developed in [LR90]. e e f This example shows that the set of k-summable series for all k > 0 is insufficient to embrace all series solutions of linear differential equations. Hav- ing in mind to sum solutions of linear differential equations another natural question is the following: Is it possible to find a (in some way, minimal) set of series endowed with a summation process compatible with the various k-summation processes, that contains all solutions of linear differential equations? From the example of the Ramis-Sibuya series we understand that any such set should contain the vector space C x of k-summable se- 0 0(1). Observe however, with the example of the 1-summable P series f(x) = x/ E(x), that not all k-summable series are solutions of linear differential equations. Since the derivative and the product of solutions of lineare differentiale equations satisfy themselves linear differential equations such a set should also contain the differential algebra Algk>0 generated by the spaces C x for all k > 0 including k =+ . And it results from the factor- { }k ∞ ization theorem of solutions of linear differential systems [Ram85], [LR94,

Thm. III.2.5] that the algebra Algk>0 suﬃces. As a homomorphism of differential algebras a summation operator on Alg , if it exists, is uniquely S k>0 determined by its values on the spaces C x generating Alg . The prob- { }k k>0 lem lies in the existence of the operator. Indeed, the compatibility condition means that the restriction of to each space C x has to be the k-summation S { }k

~~(1) One could also limit the choice to rational k > 0 since all levels of linear diﬀerential equations are rational. 174 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY~~

~~operator. Hence, given an element of Algk>0 in the form of a sum of products of k-summable series its sum is obtained by replacing each factor by its k-sum.~~

The point is that an element in Algk>0 may have several decompositions into sums of products of k-summable series and showing that these decompositions all provide the same sum is not obvious. There exists no direct proof of that fact. A solution is found in developing independently a theory of summation called multisummation which extends the k-summation processes with existence and uniqueness of sums and showing that the elements of Algk>0 are summable in that theory (cf. Prop. 8.2.14). The same question may be addressed in a given direction θ and the results are more precise. It was proved by W. Balser in [Bal92a] that, under a weak restrictive condition on levels, any multisummable series in direction θ lives in the vector space C x of k-summable series in direction θ k>0 { }k,θ for all k > 0. The decomposition is essentially unique (cf. Prop. 8.5.1) and P again the sum obtained by adding the k-sums of each term coincide with the multisum in any other usual sense whatever the decomposition. However, when the series is multisummable (i.e., multisummable in almost all direction) the decomposition depends on the chosen direction in general (cf. Sect. 8.5). One could think of this approach as a good numerical tool based only on simple summation processes. This is not the case since the decomposition into a sum is purely theoretical with no algorithm coming with. The aim of this chapter is to describe in a general setting various definitions of multisummability. Of course, we look forward to the same properties as those of k-summation, i.e., uniqueness, homomorphism of C-differential algebras,. . . . We also compare these various approaches to prove their equivalence. Comparison being not evaluation, our aim is not to grade the different approaches. None approach can be considered as being the best, none as being the worst. But any of them might be better than another one depending on the question to answer. All along the chapter we use the Ramis-Sibuya series as our reference example.

~~8.2. First approach: asymptotic deﬁnition In this section, we generalize the asymptotic approach of k-summability (cf. Sect. 6.1) to the case of several levels k~~

~~Relative Watson’s Lemma is not a parametric version of the classical Watson’s Lemma.~~

~~8.2.1. Relative Watson’s Lemma. — The Relative Watson’s Lemma is due to B. Malgrange and J.-P. Ramis [MR92].~~

Theorem 8.2.1 (Relative Watson’s Lemma).— Let 0 < k1 < k2 be given and let I be a k1-wide arc (cf. Def. 6.1.2). Then, H0 I; ≤−k1 / ≤−k2 = 0. A A When the length I of I is smaller than 2π the arc I may be supposed | | to belong to S1. Otherwise, it must be considered as an arc of the universal cover R of S1. This latter case can be reduced to the first one by an adequate ramification of the variable x. Compare Corollary 6.1.4 of Watson’s Lemma. Instead of considering a k1-exponentially flat function on a k1-wide arc I one considers here a k1- exponentially flat 0-cochain with jumps (its 1-coboundary) small enough to be k2-exponentially flat. Roughly speaking, the theorem says that the 0-cochain has too small jumps on a too large arc I to be not k2-exponentially flat itself.

In [MR92] the lemma is stated for closed k1-wide arcs. It is equivalent to choose either closed or open k1-wide arcs. Indeed, if I is closed then an element of H0 I ; ≤−k1 / ≤−k2 is represented by a 0-cochain that lives on A A a larger open arc I′. If the lemma is true for open arcs then the cochain is 0 in H0 I′ ; ≤−k1 / ≤−k2 and induces 0 in H0 I ; ≤−k1 / ≤−k2 . A A A A Conversely, suppose H0 I′ ; ≤−k1 / ≤−k2 = 0 for any closed k -wide A A 1 arc I′. Let I be an open k -wide arc and f = (f ) a 0-cochain 1 j j∈J in H0 I ; ≤−k1 / ≤−k2 associated with a covering = (I ) of I. A A I j j∈J Up to refining the covering we can assume that it is indexed by Z and I satisfies I I = if j ℓ = 1 and I I = otherwise (and thus, in j ∩ ℓ 6 ∅ | − | j ∩ ℓ ∅ particular, it has no 3-by-3 intersection), since there exists arbitrarily fine such ′ coverings of I. Write I as an increasing union of closed k1-wide sub-arcs Iℓ. Due to the form of the covering any open arc I is contained in infinitely I j many closed arcs I′; choose one of them denoted by I′ . Then, the restriction ℓ ℓj of the 0-cochain f to I′ induces 0 in H0 I′ ; ≤−k1 / ≤−k2 . This means, ℓj ℓj A A in particular, that f belongs to ≤−k2 (I ). This being true for all j Z we j A j ∈ can conclude that f induces 0 in H0 I ; ≤−k1 / ≤−k2 . A A The Relative Watson’s Lemma can be reformulated as follows. 176 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

Corollary 8.2.2.— Under the conditions of the Relative Watson’s Lemma the following natural map is injective: H0 I; / ≤−k2 H0 I; / ≤−k1 . A A −→ A A Proof. — Consider the short exact sequence 0 ≤−k1 / ≤−k2 / ≤−k2 / ≤−k1 0 −→ A A −→ A A −→ A A −→ · The associated long exact sequence of cohomology over I provides the exact sequence 0 H0(I; ≤−k1 / ≤−k2 ) H0(I; / ≤−k2 ) H0(I; / ≤−k1 ) −→ A A −→ A A −→ A A · Hence, the equivalence of the Relative Watson’s Lemma 8.2.1 and its Corol- lary 8.2.2. One can ﬁnd in [MR92] a direct proof of the Relative Watson’s Lemma (see also [Mal95]). Instead of reproducing it we prefer to include a proof of the equivalence between the Tauberian Theorem 6.3.12 and the Relative Watson’s Lemma 8.2.1 [MR92, Sect. 3 (ii)]. Lemma 8.2.3 (Malgrange-Ramis [MR92, Lemme (2.5)]) 1 Let 1/2

Proof. — The section h can be represented as a ﬁnite 0-cochain (hj)j∈J as follows. Let J = j ,j ,...,j . The components h are functions in ( ) for some { 1 2 p} j A j open sectors = I ]0,r [ with vertex 0; the 2-by-2 intersections of these j j× j sectors satisfy the conditions = and = when j ℓ > 1 j ∩ j+1 6 ∅ j ∩ ℓ ∅ | − | and we assume that the global arc I I I is less than 2π wide, the 1 ∪ 2 ∪···∪ p union I2 Ip−1 is included in I while I1 and Ip are not. Moreover, the ∪···∪ ≤−k diﬀerences h + h belong to 2 ( ) for all j J. Complete the − j j+1 A j ∩ j+1 ∈ family ( ) into a covering = of S1 (denote also = I ]0,r [ j j∈J { j}j∈J∪K j j× j 8.2. FIRST APPROACH: ASYMPTOTIC DEFINITION 177

for j K) without 3-by-3 intersections and such that ( j∈K Ij) I = . Con- ∈ • ∪ ∩ ∅ ≤−k2 sider the 1-cocycle h=(hj,j+1)j∈J∪K of with values in deﬁned by A hj(x)+ hj+1(x) if j and j + 1 J hj,j+1(x)= − ∈ ( 0 otherwise.

From the Ramis-Sibuya Theorem 6.2.1 and shrinking the sectors j if necessary, there exist functions g (x) belonging to ( ) (cf. Not. 2.3.8) such j A1/k2 j that h = g + g for all j and j +1 in J. j,j+1 − j j+1 We obtain thus the equality h (x) g (x)= h (x) g (x) on for j − j j+1 − j+1 j ∩ j+1 all j and j +1 in J and the functions h (x) g (x) glue together into a section j − j h′(x) of H0(I; ). On another hand, by construction, the g ’s for j J K A j ∈ ∪ determine an element h′′(x) of H0 S1; / ≤−k2 and we obtain A A ′ ≤−k2 ′′ h mod + h| =(hj gj)+ gj = hj on j for all j J. A j − ∈ Hence, the result. Proposition 8.2.4.— The Tauberian Theorem (Thm. 6.3.12) and the Rel- ative Watson’s Lemma (Thm. 8.2.1) are equivalent.

Proof. — Let 0 < k1 < k2 and a closed k1-wide arc I be given. By means of a convenient ramiﬁcation we may assume that the arc I is less than 2π long 1 (which implies k1 > 1/2) and this allows us to work on S . As usually, we denote s1 = 1/k1 and s2 = 1/k2. ⊲ Show that the Relative Watson’s Lemma implies the Tauberian Theo- rem 6.3.12. Let the series f(x) be both s2-Gevrey and k1-summable with sum f(x) on a k1-wide arc I. We must prove that f(x) is also k2-summable on I. As a s2-Gevrey series ande according to Corollary6.2.2, the series f(x) can be identiﬁed to an element of H0 S1; / ≤−k2e , i.e., to a 0-cochain g =(g ) A A j where g is asymptotic to f(x) for all j and g g takes its values in e≤−k2 for j i − j A all i,j. It follows from the Ramis-Sibuya Theorem 6.2.1 that gj(x) is actually s2-Gevrey (and hence also,e s1-Gevrey) asymptotic to f(x). The 0-cochain g induces canonically an element of H0 S1; / ≤−k1 , thus characterizing A A e f(x) as a s1-Gevrey series. Since f(x) is a k1-sum off(x) on I we deduce ≤−k1 that f = g mod on I (indeed, f gj is s1-Gevrey asymptotic to 0; A − |I cf.e Prop. 2.3.17). From Corollary 8.2.2 of the Relativee Watson’s Lemma, it follows that f = g mod ≤−k2 on I which proves that f(x) is k -summable A 2 on I with k2-sum f(x). Hence, the result. e 178 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

⊲ Conversely, show that the Tauberian Theorem 6.3.12 implies the Rela- tive Watson’s Lemma. Let h(x) belong to H0 I; / ≤−k2 . The section h(x) A A admits a canonical image in H0 I; / ≤−k1 : we specify h(x) mod ≤−k2 A A A when h(x) is seen as an element of H0 I; / ≤−k2 and h(x) mod ≤−k1 to A A A denote its canonical image in H0 I; / ≤−k1 . We must prove (cf. Cor. 8.2.2) A A that h(x)=0 mod ≤−k1 on I implies h(x)=0 mod ≤−k2 on I. A A From Lemma 8.2.3 there exists h′ H0(I; ) and h′′ H0 S1; / ≤−k2 ∈ A ∈ A A such that h = h′ mod ≤−k2 + h′′ . By definition, h′′(x) can be seen as A |I a 0-cochain h′′ =(h′′) where the various components h′′ are asymptotic to a j j ′′ ≤−k2 same Taylor series h (x) and hi hj takes its values in for all i,j. And − A ′′ we know, from the Ramis-Sibuya Theorem 6.2.1 that the hj (x)’s are actually ′′e s2-asymptotic to h (x). The assumption h(x)=0mod ≤−k1 on I implies h′′(x)= h′(x) mod ≤−k1 A − A on I and the seriese h′′(x) is k -summable on I with k -sum h′(x) (h′ is 1 1 − a true function; see Def.6.2.4). On another hand, by definition, h′′(x) is ′′ a s2-Gevrey series.e By the Tauberian Theorem 6.3.12 the series h (x) is then k -summable on I with the same sum h′(x). This implies thee equal- 2 − ity h′′(x)= h′(x) mod ≤−k2 . Hence, h(x)=0 mod ≤−k2 as followed.e − A A 8.2.2. Asymptotic definition of multisummablilty. — Towards the generalization of the asymptotic definition (cf. Def. 6.1.6, Sect. 6.1) of k- summability we proceed as follows. Begin with the case of two levels k1

~~letting f1 understood. In the latter case, one also says that f2 isa(k1,k2)-sum of f(x) in direction θ.~~

Remarke 8.2.6. — Suppose I1 and I2 are closed arcs. Sections over I1 or I2 live then on larger open arcs. From Condition (ii), one can represent f1 by a 0-cochain containing f2 as a component. One can also choose a 0-cochain over an open covering of I1 by arcs with no 3-by-3 intersection and no intersection 2-by-2 on I2. Thus, Definition 8.2.5 can be reformulated as follows: The series f(x) is (k1,k2)-summable on (I1,I2) if there exists a 0- cochain f, s1-Gevrey asymptotic to f(x) on I1, that has no jump on I2 and only k -exponentiallye flat jumps on I I . 2 1 \ 2 The couple (f ,f ) where f is thee natural image of f in H0(I ; / ≤−k2 ) 1 2 1 1 As1 A and f2 its restriction to I2 is a (k1,k2)-sum of f(x) on (I1,I2). Recall that, in general, a 0-cochain which is s1-Gevrey asymptotic to a given series may have jumps (its coboundary)e as large as k1-exponentially flat (Prop. 2.3.17). The condition that the jumps are k2-exponentially flat is strong and guaranties the uniqueness of the (k1,k2)-sum of f(x) on (I1,I2) as we show below. e Watson’s Lemma and the Relative Watson’s Lemma give sense to the notion of (k1,k2)-summability by implying the uniqueness of (k1,k2)-sums.

~~Proposition 8.2.7 (Uniqueness of (k1 k2)-sums).— The multisum (f1,f2) of f(x) on (I1,I2), when it exists, is unique.~~

′ ′ Proof. — Suppose (f1,f2)e and (f1,f2) are (k1,k2)-sums of f(x) on (I1,I2). By Prop. 2.3.17 the diﬀerence f f ′ belongs to H0 I ; ≤−k1 / ≤−k2 and 1 − 1 1 A A the Relative Watson’s Lemma (Thm. 8.2.1) implies that f ef ′ = 0. This, in 1 1 ′ ≤−k2 − turn, implies that f2 = f2 mod and from the classical Watson’s Lemma ′ A (Thm. 6.1.3) that f2 = f2 since I2 is k2-wide.

~~Example 8.2.8.— The Ramis-Sibuya series RS(x) (Exa. refRSseries) is (1, 2)- summable on (I1,I2) if and only if I1 does not contain the Stokes arc of the Eu-~~

ler series [π/2, 3π/2] and I2 contains none of the twof Stokes arcs of the Leroy series [π/4, 3π/4] mod π. Make explicit the (1, 2)-summability of RS(x) according to Deﬁnition 8.2.5 above in

the case when, for instance, I1 = [0,π] and I2 = [0,π/2] I1. Choose 0 <ε<π/4 and ′ ⊂ ′ consider the open covering of I1 by the arcsf I1 =]π/2,π + ε[ and I2 =] ε,π/2+ ε[. ′ I − Notice that I I2 = as mentioned to be a possible choice in Remark 8.2.6. Denote 1 ∩ ∅ temporarily by E(x) the determination of the Euler function deﬁned on ε < arg(x) < ′ ′ ′ ′ − π + ε. Denote by E1 and E2 the restrictions of E to I1 and I2 respectively. Clearly, the 180 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

′ ′ ′ ′ 0-cochain (E1, E2) of takes its values in 1 and has a trivial coboundary E1 E2 0. ′ ′ I A ′ ′ − ′ ≡ ′ Denote by L1 and L2 the 2-sums of L(x) on I1 and I2 respectively. The 0-cochain (L1, L2) ′ ′ of takes its values in 1/2 hence also in 1 and its coboundary L1 L2 belongs to 0I ′ ′ ≤−2 A A ′ ′ ′ ′ − H I1 I2; . It follows that the 0-cochain (E1 +L1, E2 +L2) determines an element ∩0 A ≤−2 0 ′ ′ f1 of H I1; 1/ . Denote by f2 the element of H (I2; 1) deﬁned by E +L . Both A A A 2 2 f1 and f2 are 1-asymptotic to the Ramis-Sibuya series RS(x). The couple (f1,f2) is the (1, 2)-sum of RS(x) on (I1,I2).

~~Conversely, if I1 contained the Stokes arc [π/2, 3π/f2] of the Euler series E(x) then~~

we would havef to use two different determinations of E on I1 generating a non trivial coboundary with values in ≤−1 and not in ≤−2 and, thus, Condition (i)e of Defini- A A tion 8.2.5 would fail. If I2 contained a Stokes arc [π/4, 3π/4] or [ 3π/4, π/4] of the ′ − − Leroy series L(x) then we would have to split the 2-sum L2 of L(x) into a 0-cochain with non trivial coboundary on I2 and Condition (ii) of Definition 8.2.5 would fail. e e Let us now state the general case.

~~Deﬁnition 8.2.9 (multi-level k and k-multi-arc).—~~

~~⊲ We call multi-level, and we denote by k = (k1,k2,...,kν), any ﬁnite sequence of numbers k1,k2,...,kν satisfying the conditions~~

~~Observe that our levels are always ordered in increasing order: k~~

~~Deﬁnition 8.2.10 (multisummability).—~~

A series f(x) is said to be k-summable on I with k-sum f =(f1,f2,...,fν) if 0 ≤−kj+1 ⊲ fj belongs to H Ij; s1 / for all j = 1, 2,...,ν 1; e A A − ⊲ f belongs to H0 I ; ; ν j As1 ⊲ the fj’s are compatible:

~~≤−kj+1 fj| = fj+1 mod for all j = 1, 2,...,ν 1; Ij+1 A − 8.2. FIRST APPROACH: ASYMPTOTIC DEFINITION 181~~

~~⊲ for all j, the section fj is s1-Gevrey asymptotic to the series f(x) on Ij:~~

~~Ts1,Ij fj(x)= f(x) for all j = 1, 2,...,ν. e~~

In the case when the arcs I1,Ie2,...,Iν are all bisected by a same direction θ then f is called a k-sum of f(x) in direction θ. By abuse of language, one sometimes talks of fν as a k-sum of f(x) on I, the components f1,f2,...,fν−1 being understood. e e Remark 8.2.11. — Remark 8.2.6 can be generalized as follows. Suppose I is a closed k-multi-arc. Deﬁnition 8.2.10 is equivalent to saying that

~~f(x) is k-summable on I with k-sum f =(f1,f2,...,fν) if there exists a~~

0-cochain f on I1 which is s1-Gevrey asymptotic to f(x), which has no e jump on Iν and otherwise, has jumps that are at least kν-exponentially flat on Iν−1, kν−1-exponentially flat on Iν−2,... and ke2-exponentially flat on I1. If so, the k-sum f is defined by taking as f , for j = 1, 2,...,ν 1, j − the natural image of f in H0 I ; / ≤−kj+1 and taking as f the j As1 A ν restriction of f to I . ν Proposition 8.2.12 (uniqueness).— The k-sum f(x), when it exists, is unique.

Proof. — The proof proceeds as in the case of two levels (cf. Prop. 8.2.7). ′ ′ ′ Suppose (f1,f2,...,fν) and (f1,f2,...,fν) are k-sums of f(x) on I. By Proposition 2.3.17 the diﬀerence f f ′ belongs to H0 I ; ≤−k1 / ≤−k2 1 − 1 1 A A and, I being k -wide, the Relative Watson’s Lemma (ecf. Thm. 8.2.1) 1 1 implies that f f ′ =0. This, in turn, implies that f f ′ belongs to 1 − 1 2 − 2 H0 I ; ≤−k2 / ≤−k3 . Again, the Relative Watson’s Theorem implies 2 A A that f f ′ = 0 and that f f ′ belongs to H0 I ; ≤−k3 / ≤−k4 . And 2 − 2 3 − 3 3 A A so on, until the νth step where f f ′ belongs to H0 I ; ≤−kν and we ν − ν ν A conclude by the classical Watson’s Lemma (Thm. 6.1.3) that f = f ′ since I ν ν ν is kν-wide.

Notation 8.2.13 (multisummable series).— Given a multi-level k and a k-multi-arc I we denote by ⊲ C x the set of k-summable series on I; { }{k,I} ⊲ C x the set of k-summable series in direction θ. { }{k,θ} 182 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

⊲ Sum the subset of ν−1 H0 I ; / ≤−kj+1 H0 I ; made {k,I} j=1 j As1 A × ν As1 of the elements satisfying the compatibility condition of Deﬁnition 8.2.10 (iii). Q ⊲ : C x Sum the k–summation operator on I which S{k,I} { }{k,I} −→ {k,I} to any k-summable series on I associates its unique k-sum on I according to Proposition 8.2.12. ⊲ Sum and instead of Sum and in direction θ. {k,θ} S{k,θ} {k,I} S{k,I} We leave as an exercise the proof of the following proposition generalizing Proposition 6.1.10.

Proposition 8.2.14.— Let k =(k1,k2,...,kν) and I =(I1,I2,...,Iν) be a multi-level and a k-multi-arc (cf. Def. 8.2.9). (i) The set C x is a differential sub-algebra of the differential C-al- { }{k,I} gebra C[[x]]s1 of s1-Gevrey series. (ii) Let k′ be a multi-level extracted from k and I′ the corresponding k′- multi-arc extracted from I. Then, C x ′ ′ is a differential sub-algebra of C x . { }{k ,I } { }{k,I} In particular, the differential algebras C x of k -summable series { }{kj ,Ij } j on I for j = 1, 2,...,ν are differential sub-algebras of C x . j { }{k,I} (iii) The Taylor map T : Sum C x s1,I {k,I} −→ { }{k,I} is an isomorphism of differential C-algebras with inverse the k-summation operator on I. Sk,I Remark 8.2.15. — The previous proposition asserts that C x contains { }{k,I} the differential algebra generated by the algebras C x ,j = 1, 2,...,ν of { }{kj ,Ij } kj-summable series on Ij. It will be shown in Section 8.5 that the two algebras are actually equal. Although we do not provide an extensive proof of Proposition 8.2.14 let us observe how a kj-summable series may be regarded as a k-summable series. Consider the example of j = ν, all cases being similar. Suppose f(x) is kν-summable on Iν. This means that there exists a function (its kν-sum) f H0 I ; satisfying T f (x)= f(x). This implies also thate the ν ∈ ν Asν sν ,Iν ν series f(x) is sν-Gevrey and the Borel-Ritt Theorem (Cor. 2.4.4) allows to ′ e complete the sum fν into a 0-cochain fν over I1 whose components are all sν- Gevreye asymptotic (hence, s1-Gevrey asymptotic) to f(x) and its coboundary has values in ≤−kν . Recall that the sheaves ≤−k satisfies the in- A A clusions ≤−kν ≤−kν−1 ≤−k1 . Thus, thee 0-cochain f ′ induces A ⊂A ⊂···⊂A ν 8.3. SECOND APPROACH: MALGRANGE-RAMIS DEFINITION 183 canonically elements f H0 I ; / ≤−k1 , f H0 I ; / ≤−k2 and 1 ∈ 1 As1 A 2 ∈ 2 As1 A so on... and f H0 I ; defined on I ,I ,...,I respectively and satis- ν ∈ ν As1 1 2 ν fying Def. 8.2.10. Thus, (f1,f2,...,fν) defines the k-sum of f(x) on I. To end this section let us mention the fact that Propositione 6.1.11 and its Corollary 6.1.12 remain valid if one replaces k-summability by multisummability. Recall notations of Sections 2.3.2 and 6.1: given a series g(t) we denote by gj its r-rank reduced series defined for j = 0, 1,...,r 1, by g(t)= r−1 tjg (tr); − j=0 j given an arc I =(α,β)we denote by Iℓ the arc (α + 2ℓπe)/r, (β + 2ℓπ)/r .e /r P We can state: e e Proposition 8.2.16.— Let r> 1 be an integer.

: (i) Extension of the variable. A series f(x) is k-summable on I =(I1,I2,...,Iν) r if and only if the series g(x) = f(x ) is rk-summable on I/r = (I1/r,I2/r,...,Iν/r). e e e ℓ : (ii) Rank reduction. The series g is rk-summable on the arcs I/r for all ℓ = 0, 1,...,r 1 if and only if the series g for j = 0, 1,...,r 1, − j − are k-summable on I. e e Proof. — (i) Let f(x)=(f1(x),f2(x),...,fν(x)) be the k-sum of f(x) on . r r r I Then, g(x)=(g1(x) = f1(x ),g2(x) = f2(x ),...,gν(x) = fν(x )) is the rk- sum of g(x) on . e I/r (ii) Suppose the series gj(x) are all k-summable on I. By deﬁnition, these e r−1 j r series satisfy g(t)= j=0 t gj(t ). From (i) and Proposition 8.2.14 it follows that g(t) is rk-summable one the arcs corresponding to the various determi- 1/r P j r r−1 ℓ(r−j) ℓ nations of t =e x . Conversely,e use formula rt gj(t )= ℓ=0 ω g(ω t) 2πi/r r 0 wheree ω = e to conclude that the series gj(t ) are rk-summable on I P /r and then, using (i) again, the series gj(x) are k-summablee on I. e Proposition 8.2.16 allows us to assume k largee or small at convenience. In what follows, the assumption k > 1/e2 is quite often convenient.

~~8.3. Second approach: Malgrange-Ramis deﬁnition~~

For convenience, we assume any k > 1/2 so that arcs of length π/k are proper sub-arcs of the circle S1. This is always made possible by a convenient change of variable x = tr according to Proposition 8.2.16. Like in the 184 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

~~Ramis-Sibuya approach for k-summability, the aim is now to get rid of Gevrey asymptotics.~~

8.3.1. Deﬁnition. — Suppose we are given an s1-Gevrey series f(x), a multi-level k = (k1,k2,...,kν) and a k-multi-arc I = (I1,I2,...,Iν). Denote by ϕ H0 S1; / ≤−k1 the k -quasi sum of f(x). e 0 ∈ A A 1 Deﬁnition 8.3.1 (Malgrange-Ramis multisummability) e The series f(x) is said to be k-summable on I with k-sum f =(f1,f2,...,fν) if ⊲ f ebelongs to H0 I ; / ≤−kj+1 for all j = 1, 2,...,ν 1; j j A A − 0 ⊲ fν belongs to H Iν; ; A ⊲ the fj’s are compatible:

~~≤−kj+1 fj| = fj+1 mod for all j = 1, 2,...,ν 1; Ij+1 A − ⊲ the fj’s are compatible with the k1-quasi-sum ϕ0 of f(x):~~

≤−k1 ϕ0| = fj mod for all j = 1, 2,...,ν. Ij A e Proposition 8.3.2.— Deﬁnitions 8.2.10 and 8.3.1 of multisummability are equivalent (with same sums). In particular, Malgrange-Ramis k-sum, when it exists, is unique.

Proof. — Suppose f =(f1,f2,...,fν) is a k =(k1,k2,...,kν)-sum of f(x) on I in the sense of Def. 8.3.1. By the Ramis-Sibuya Theorem 6.2.1 the k1-quasi- sum ϕ of f(x) belongs to H0 S1; / ≤−k1 and is asymptotic toe f(x). 0 As1 A The compatibility Condition (iv) implies that the same is true for f; hence, Definition 8.2.10e is satisfied. e Conversely, suppose that f is a k-sum of f(x) on I following Definition 8.2.10. By the Borel-Ritt Theorem 2.4.1 a 0-cochain representing f1 can be 1 completed into a 0-cochain over S representinge ϕ0; hence, satisfying Defini- tion 8.3.1.

~~Example 8.3.3.— Let us go back to Example 8.2.8 in view to make explicit Malgrange-Ramis deﬁnition for the Ramis-Sibuya series RS(x) and prove its (1,2)-~~

~~summability on the (1,2)-multi-arc I = (I1,I2) where I1 = [0,π] and I2 = [0,π/2] I1. ⊂ We want to represent the 1-quasi-sum of RS(x) by a 0-cochainf with no jump on I2, ﬂat~~

jumps of exponential order at most 2 on I1 and ﬂat jumps of order at most 1 out of ′ 1 I1. To this end, choosing again 0 <ε<π/f 4, we consider the open covering of S I by I′ =]π/2, 3π/2[, I′ =] ε,π/2+ ε[ and I′ =] π/2 ε, 0[. Denote by E+ the deter- 1 2 − 3 − − mination of the Euler function E(x) deﬁned on ] π/2, 3π/2[ and by E− its determination − 8.3. SECOND APPROACH: MALGRANGE-RAMIS DEFINITION 185

on ] 3π/2,π/2[. Denote by L+ the 2-sum of the Leroy series L(x) on ] 3π/4, 3π/4[ and − − by L− its 2-sum on ]π/4, 7π/4[. e ′ The 1-quasi-sum ϕ0 of RS(x) is represented by the 0-cochain of deﬁned as follows: I RS (x)=E+(x)+L−(x) on I′ f 1 1 + + ′  RS2(x)=E (x)+L (x) on I2  − + ′  RS3(x)=E (x)+L (x) on I3. 

~~Figure 2~~

− + 1/x2 We observe that RS1 RS2 = L L = ce (c is the corresponding Stokes − − ′ ′ multiplier) is exponentially ﬂat of order 2 on I1 I2 while RS2 RS3 and RS3 RS1 are ′ ′ ′ ∩′ − − exponentially ﬂat of order 1 on I I and I I respectively. The (1,2)-sum (f1,f2) of 2 ∩ 3 3 ∩ 1 RS(x) on (I1,I2) is given by the restriction of ϕ0 to I1 and I2 respectively, i.e., , for f1 by

the 0-cochain (RS1, RS2) and for f2 by f2 = RS2. f 8.3.2. Application to differential equations. — In this section, we extend Theorem 6.2.5 to the case of several levels. We consider a linear differential equation (or system) Dy = 0 with meromorphic coefficients at 0 and we suppose that the equation Dy = 0 has a series solution f(x) with multi-level k = (k1,k2,...,kν) (cf. Def. 4.3.6 (iv)). Recall that levels are ordered in increasing order: k1

~~Proof. — Recall that with no loss of generality we assume that k1 > 1/2. 186 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY~~

The Stokes phenomenon says that solutions asymptotic to f(x) may be continued around 0 as long as no Stokes arc is passed. When a Stokes arc of one or several levels kℓ is passed to stay asymptotic to f(x) the solutione must be added kℓ-exponentially flat terms. Thus, an asymptotic solution on Iν, given, for instance, by the Main Asymptotic Existence Theoreme 4.4.2 may be continued by adding kν-exponentially flat terms over Iν−1, kν−1-exponentially flat terms on Iν−2, ..., k2-exponentially flat terms on I1. Outside of I1 it can 1 be continued into a full 0-cochain over S by allowing jumps k1-exponentially small. Hence, a k-sum of f(x) on I.

e 8.4. Third approach: iterated Laplace integrals The method is due to W. Balser [Bal92b]. It proceeds by recursion and is based on the fact that a convenient Borel transform of the series is itself summable with conditions that we make explicit below. Among the known ones this approach is probably the best from a numerical view point to numerically evaluate multisums.

~~As previously, we develop ﬁrst the case of two levels (k1,k2). Suppose we are given 0~~

Denote by θ1,θ2 the bisecting directions of I1,I2 and by I1, I2 arcs centered at θ ,θ with length I = I π/k π/κ and I = I π/k respec- 1 2 | 1| | 1|− 2 ≥ 1 | 2| | 2|− 2 tively. We assume that I1 and I1, resp. I2 and I2, are simultaneouslyb b open or closed(2). b b b b Deﬁnition 8.4.1 ( (k1 k2)-Li-summability).— A series f(x) is said to be (k1,k2)-summable by Laplace iteration on (I1,I2) (in short, (k1,k2)-Li- summable on (I ,I )) if its k -Borel transform g(ξ) = (f)(eξ) satisﬁes the 1 2 2 Bk2 following two conditions: e ⊲ g(ξ) is κ1-summable on I1 b

b b (2) In case I is closed of size |I | = π/k then I reduces to one point. Recall that, handling 2 2 2 b2 presheaves or sheaves, asymptotic conditions of sections over a closed set are valid on a convenient larger open set. 8.4. THIRD APPROACH: ITERATED LAPLACE INTEGRALS 187

⊲ its κ1-sum g(ξ) can be analytically continued to an unlimited open sector Σ containing I ]0, + [ with exponential growth of order k at inﬁnity. 2× ∞ 2 The (k1,k2)-Li-sum f(x) of f(x) on (I1,I2) is deﬁned as b Li f (xe)= k2,θ(g)(x) L for all direction θ Σ and corresponding x. ∈

~~It follows from the deﬁnition, that the (k1,k2)-Li-sum of f(x) on (I1,I2), when it exists, is unique. e~~

~~Example 8.4.2.— (even part of the Ramis-Sibuya series) Again, we illustrate the deﬁnition on the example of the Ramis-Sibuya series RS(x)~~

~~(Exa. 8.1.1) for which k = (k1,k2) = (1, 2) and, again, we choose I = (I1,I2) with~~

I1 = [0,π] and I2 = [0,π/2] I1. Then, I1 =[π/4, 3π/4], I2 = π/4 and κ1 = 2 sof that ⊂ { } we must apply to RS(x) a 2-Borel transform. In order to make the calculations simpler we choose to use rank reduction of orderb two, i.e., to performb the calculations separately on the even and thef odd part of the series. Treat now the case of the even part and thus, 0 0 consider the series M(x) = E (x)+ L(x) where E (x) = (2n 1)! x2n is the even n≥1 − − part of the Euler series E(x). P e e e 0 e 0 Look first at whate happens to E (x) after a 2-Borel transform. The series E (x) satisfies the equation e e (53) Dy x4y′′ +2x3y′ y = x2. ≡ − The homogeneous equation Dy = 0 admits the two linearly independent solutions e1/x 0 and e−1/x. It follows that the anti-Stokes directions for E (x) are the two real half-axis. To apply a 2-Borel transform one has to apply a ramification x = t1/2 followed by a 1-Borel 0 transform and the inverse ramification. So, set x = t1/2.e The series E (t1/2) is a series in integer powers of t this is why we separated the even and the odd part of RS(x) and it satisfies the equation e 4t3Y ′′ +6t2Y ′ Y = t. f − Its 1-Borel transform Y (τ) satisfies the equation 4τ 2Y ′ + (6τ 1)Y =1 b − (Substitute τ for t2d/dt and d/dτ for 1/t. If we had not restricted the study to an b b even series, terms in t1/2 would appear and this Borel equation would be a much more complicated convolution equation). With the inverse ramification τ = ξ2, the formal 0 2-Borel transform U(ξ)= Y (τ), of E (x), satisfies the equation (54) ∆U 2ξ3U ′ + (6ξ2 1)U =1. b b ≡e − The Newton polygon of ∆ has a unique slope, equal to 2, at 0 and a null slope at b b b infinity. 188 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

~~Figure 3. Newton polygons of D (slope 1) and ∆ (slope 2)~~

The 1-dimensional space of solutions of the homogeneous equation ∆U = 0 is gener- 2 ated by u(ξ)= e−1/(4ξ )/ξ3. The anti-Stokes directions of equation (54) are then the two real half-axis and, in all other direction θ, the series U(ξ) is 2-summable (cf.b Thm. 6.2.5).

For the choiceb θ = π/2 this means that U(ξ) is 2-summable on I1. Hence, the first condition of Definition 8.4.1 is satisfied. On another hand,b the 2-sum U(ξ) of U(ξ) satisfies equation (54) which admits 0 and infinityb as unique singular points.b It can then be analytically continued up to infinity in the direction θ = π/4 and neighboringb ones, i.e., to

an unlimited open sector Σ containing I2 ]0, + [. From the Newton polygon of ∆ we × ∞ know that all solutions of (54) have moderate growth at infinity and the second condition b 0 of Definition 8.4.1 is satisfied. It follows that the 2-Laplace transforms Eθ(x) of U(ξ) are defined in all direction θ Σ. The functions E0(x) are analytic continuation from each ∈ θ others since U(ξ) admits no singular point in these directions and therefore, they define 0 0 the (1, 2)-Li-sum E (x) of E (x) on (I1,I2). To summarize:

0 0 E (x)= 2,θ 2,θ 2,θ 2,θ E (x) θe L ◦L ◦B ◦B where the Borel series are replaced by their sum and analytic conti nuation when necessary. 0 e0 Notice that the function Eθ(x), although asymptotic to E (x), is not 1/2-Gevrey asymp- 0 totic since, otherwise, the series E (x) would be a 1/2-Gevrey series (cf. 2.3.10), which is not. e Look what happens to thee series L(x) in this procedure. The 2-Borel transform n 2n 2 of L(x) produces the convergent series V (ξ) = n≥0( 1) ξ = 1/(1 + ξ ). It is then, e − 2-summable in all direction, and especially on I1P. It can be continued up to inﬁnity with moderatee growth but in the directions θ = π/2. It can then be applied a 2-Laplace ± transform in any direction of an unlimited openb sector Σ containing I2 ]0, + [ to deﬁne × ∞ the (1,2)-Li-sum L(x) of L(x) on (I1,I2).

We conclude that the series M(x) is (1,2)-Li-summable on (I1,I2b). e Compare (1,2)-sum and (1,2)-Li-sum. Denote by M Li(x) the (1, 2)-sum of M(x) 0 e 0 on (I1,I2). The (1,2)-Li-sum E (x) of E (x) can be continued all over I1 by applying 2-Laplace transforms in directions θ from π/4 ε to 3π/4+ ε (indeed, the uniquee anti- − Stokes directions for U(ξ) are θ = 0 ande θ = π; therefore, the 2-sum U(x) of U(ξ) exists. It can be continued with moderate growth at inﬁnity on the unlimited sector π/4 < arg(ξ) < 5π/b 4). Taking 2-Laplace transforms in directions θ ] 3π/4, bπ/4[ − ∈ − − allows to complete E0(x) into an element 0 of H0 S1; / ≤−1 . Similarly, the section L E 0 1A A ≤−k2 0 over I2 can be completed into an element H S ; / and the sum + L∈ A A E L 8.4. THIRD APPROACH: ITERATED LAPLACE INTEGRALS 189

~~1 provides a k1-quasi-sum of M(x) in the form of a section over S with no jump on I2 and~~

~~ﬂat jumps of exponential order 2 on I1 and of order 1 out of I1, so that, Deﬁnition 8.3.1 is~~

satisfied. By restriction to I1eand I2, this k1-quasi-sum determines the (1, 2)-sum (M1,M2) Li of M(x) on (I1,I2) in the sense of Definition 8.3.1. It follows that M2(x)= M (x) on I2. This fact is general as it is proved below (Balser-Tougeron Theorem). e Let us finally observe that the previous procedure can as well be applied to the odd part of RS(x) after factoring x. This shows that the procedure applies to the Ramis-Sibuya series RS(x) itself giving rise to the same (1, 2)-sum as before. f Thef main result is as follows. Theorem 8.4.3 (Balser-Tougeron: case of two levels)

(k1,k2)-Li-summability on (I1,I2) and (k1,k2)-summability on (I1,I2) are equivalent with “same” sum. Li Precisely, if f denotes the (k1,k2)-Li-sum of a series f(x) and (f1,f2) Li its (k1,k2)-sum on (I1,I2) then, f2(x)= f (x) on I2 and thus, e ′ ′ f2(x)= k ,θ κ ,θ κ1 k f(x) L 2 2 ◦L 1 1 ◦B ◦B 2 ◦ when the formula makes sense and especially, for directions θ′ and θ′ close e 1 2 enough to the bisecting direction θ2 of I2. The latter formula explains the denomination “by Laplace iteration”.

~~Proof. — For simplicity of language assume that I1 and I2 are closed arcs.~~

⊲ Prove that (k1,k2)-summability implies (k1,k2)-Li-summability. We use the notations of Definition 8.4.1 and above. In particular, κ1 is given by the formula 1/κ = 1/k 1/k . By hypothesis, there exists 1 1 − 2 a k1-quasi-sum of f(x) which induces the function f2(x) on I2 and the 0-cochain f1(x) on I1 (using the same notation for the 0-cochain and the element of H0 I ; e/ ≤−k2 it defines); the coboundary of f has values in 1 A A 1 ≤−k2 (no jump allowed on I and exponentially flat jumps of order at most A 2 k on I I ; cf. Def. 8.3.1). Denote again g(ξ)= (f)(ξ). 2 1 \ 2 Bk2 For simplicity, we assume that f1(x) has only one jump, the case of more jumps being treated similarly. Thus, assumeb that, in restrictione to I1 (i.e., to a neighborhood of I1), the k1-quasi-sum reduces to two components: f(x) = ∗ ∗ f2(x) over an open arc I containing I2 and f (x) over an open arc I which we can assume to satisfy I∗ I = jointly with I I∗ I . ∩ 2 ∅ ∪ ⊃ 1 The proof of Theorem 6.3.11 (part (i) = (ii)) remains valid for f(x) ⇒ and I I although I is shorter than π/k . Like in Theorem 6.3.11, denote ⊃ 2 1 190 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

by I the (open since I is open) arc deduced from I by shortening it of π/(2k2) on both sides. It follows that there exists an unlimited sector Σ containing I ]0b, + [ on which there is an analytic function g(ξ) both κ -Gevrey asymp- × ∞ 1 totic to g(ξ) at 0 and having exponential growth of order at most k2 at inﬁnity. Asb I is smaller than I the sector Σ has opening smaller than I = π/κ but 1 | 1| 1 containsb the direction θ2. The question is to analytically continue g(ξ) into a κ1-Gevrey asymptotic function on I1. To this end, let us use againb the variables Z = 1/xk2 , ζ = ξk2 and notations as in the proof of Theorem 6.3.11 choosing θ2 = 0 by means of a rotation.b In particular, the series f(x) reads now F (Z)= a /Zn/k2 . We are led to the following situation: n≥k0 n e : ⊲ a functionP F (Z) satisfying the asymptotic Condition (42) at inﬁnity on ae sector =[ ω , +ω ] [R , + [ − 1 2 × 0 ∞ which contains the right half-plane (Z) > 0 but the disc Z 0 such that

~~N−1 a AN F (Z) n CN N/k1 for all N and all Z . − Zn/k2 ≤ Z N/k2 ∈ n=k0 | | X~~

We also assume that π < ω and ω∗ < ω < ω∗ < π. Otherwise, we − − 1 1 2 2 would proceed in several steps like in the proof of Theorem 6.3.11. Deﬁning G(ζ) by G(ζ) = 1/(2πi) F (U)eζU dU where γ = γ γ γ γ 1 ∪ 2 ∪ 3 (see Fig 4) is the boundary of provides a function satisfying Condition R (ii) of Theorem 6.3.11 on the sector Σ = ω + π/2 < arg(ζ) < +ω π/2 {− 2 1 − } (condition translated in the variable ζ = ξk2 ). Observe that Σ contains the direction θ2 = 0. Extend now the domain of deﬁnition of G(ζ) by moving γ1 ∗ ∗ towards γ1 in the direction ω2. To this end, set

~~1 G∗(ζ)= F ∗(U)eζU dU + F (U)eζU dU . 2πi γ∗ γ ∪γ Z 1 Z 2 3 8.4. THIRD APPROACH: ITERATED LAPLACE INTEGRALS 191~~

~~Figure 4~~

By Cauchy’s Theorem the difference ∆(ζ)= G(ζ) G∗(ζ) is given by − 1 ∆(ζ) = F (U) F ∗(U) eζU dU 2πi γ1 − iω Z +∞ e 2 iω = F (Veiω2 ) F ∗(Veiω2 ) eζVe 2 dV. 2πi − ZR0 However, by hypothesis, there exists A > 0 such that F (U) F ∗(U) −A|U| | − | ≤ e (the coboundary in the variable x is exponentially flat of order k2) so that iω iω F (Veiω2 ) F ∗(Veiω2 ) eζVe 2 e −A+ℜ(ζe 2 ) V | − | ≤ and the function ∆(ζ) is defined and holomorphic on the half-plane (ζeiω2 ) < ℜ A. The function G∗(ζ)+∆(ζ) provides the analytic continuation of G(ζ) to a (limited) sector Σ∗ based on I∗ =[ ω∗+π/2, ω∗+π/2] near 0. From now on, − 2 − 1 denote by G(ζ) this analytic continuation of the initial G(ζ)toΣ∗ Σ. Observe ∪ that the arguments above areb unable to prove that the analytic continuation on Σ∗ can be pushed up to infinity. This might however be possible, for instance, by Theorem 6.3.11, when the series f(x) is not only (k1,k2)-summable on (I1,I2) but k1-summable on I1. Recall Condition (41) for G(ζ) one Σ: there exist A′,C′ > 0 such that

N−1 a G(ζ) n ζn/k2−1 C′N N/κ1 A′N ζ N/k2−1 for all N and all ζ Σ. − Γ(n/k2) ≤ | | ∈ n=k0 X Like the initial one the new function G(ζ) satisfy an asymptotic condition of the same type as Condition (41) at 0 on Σ Σ∗ since this is the case for G(ζ) ∪ on Σ and for G∗(ζ) and ∆(ζ) on Σ∗. We can thus conclude that g(ξ)= G(ξk2 ) 192 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

~~is κ1-asymptotic to g(ξ) on I1. This achieves the proof of the fact that f(x) is (k1,k2)-Li-summable on (I1,I2). e b e~~

⊲ Prove that (k1,k2)-Li-summability implies (k1,k2)-summability. Conversely, suppose G(ζ) is defined on Σ Σ∗, has exponential growth at ∪ infinity on Σ and satisfies at 0 the asymptotic Condition (41) on Σ Σ∗. ∪ From the proof of Theorem 6.3.11 we know that Laplace transforms in directions belonging to Σ provide a function F (Z) which satisfies Condition (42) on [ ω , ω ]. − 1 2 Let b = b eiβ Σ Σ∗ with, say, β = ω + π/2, and d be the half-line | | ∈ ∩ − 2 β issuying from b in direction β. Consider the truncated Laplace transform b b −Zζ F (Z)= 0 G(ζ)e dζ. We prove, like for P1 in the proof of Theorem 6.3.11, that F b(Z) satisfies an estimate of the type of (42) on a half-plane (Zeiβ) > 0 R ℜ (with new constants). Since G(ζ) has exponential growth G(ζ) C′′eB|ζ| b −Zζ | | ≤ on dβ the difference F (Z) F (Z)= G(ζ)e dζ satisfies − db +∞ R iβ F (Z) F b(Z) C′′ e(B−ℜ(Ze ))τ dτ | − | ≤ Z|b| and has then exponential decay on the half-plane (Zeiβ) > B based on the ℜ arc ]ω2 π, ω2[ (i.e., bisected by β = ω2 π/2). − ∗ iβ∗ − ∗ −∗ ∗ Consider now b = b e with β = ω2 + π/2 in Σ . ∗ b∗ | | b −Zζ− The function F (Z)= 0 G(ζ)e dζ satisfies an estimate of type (51) ∗ ∗ on the half-plane (Zeiβ ) > 0 and the difference F b(Z) F b (Z) = −Zζ ℜ R − ⌢ G(ζ)e dζ is exponentially small of order one on the intersection β∗β ∗ Rof the two half-planes (Zeiβ) > 0 and (Zeiβ ) > 0 (since G(ζ) is bounded ⌢ ℜ ℜ on the arc β∗β). Turning back to the variable x this provides functions ∗ ∗ k2 b b k2 f(x) = F (x ) and f (x) = F (x ) that are k1-Gevrey asymptotic to ∗ ∗ f(x) on I I2 and I I1 I2 respectively, the difference f(x) f (x) ⊃ ⊃ \ ∗ − being exponentially flat of order k2 on I I . In other words, the couple ∗ ∗ b b k2 k2 ∩ (ef (x)= F (x ),f(x)= F (x )) defines a (k1,k2)-sum of f(x) on (I1,I2).

e ⊲ Prove the formula. We have proved that the (k1,k2)-Li-sum and the ﬁnal (k1,k2)-sum are both equal to f(x). By construction, F (Z) is the Laplace transform of G(ζ) in directions belonging to Σ, that is, f(x) is the k2-Laplace ′ transform of g(ξ) in directions θ2 close to the bisecting line θ2 of I2. The κ1-sum ′ ′ g(ξ) of g(ξ) reads g(ξ)= κ ,θ κ1 (g)(ξ) for θ I1 and g(ξ) = k (f)(ξ) L 1 1 ◦B 1 ∈ B 2 b b b b e 8.4. THIRD APPROACH: ITERATED LAPLACE INTEGRALS 193 by deﬁnition. Hence, the result

~~′ ′ f(x)= k ,θ κ ,θ κ1 k f(x) L 2 2 ◦L 1 1 ◦B ◦B 2 ◦~~

~~′ ′ e for all compatible choices of θ1,θ2 and x. This ends the proof.~~

Summability by Laplace iteration can be generalized by induction to the case of any multi-level k =(k1,k2,...,kν) as follows. Let I = (I ,I , ,I ) be a k-wide multi-arc. Recall (cf. Def. 8.2.9) 1 2 ··· ν that this means that 0

~~b Deﬁnition 8.4.4 (summability by Laplace iteration: general case)~~

A series f(x) is said to be k-summable by Laplace iteration on I ( in short, k-Li-summable on I) if its k -Borel transform g(ξ) = (f)(ξ) satisfies the ν Bkν following twoe conditions: ⊲ g(ξ) is k-Li-summable on I, b e ⊲ its k-Li-sum g(ξ) can be analytically continued to an unlimited open sector Σbcontainingb I ]0, + [ withb exponential growth of order k at infinity. ν× ∞ ν The k-Li-sumb f(x) of f(x) on I is defined as f Li(x)= (g)(x) for all Lkν ,θ direction θ Σ andb corresponding x. ∈ e

~~From the deﬁnition, the k-Li-sum f Li when it exists is unique. Denote by κ1,κ2,...,κν the numbers given by~~

~~1 1 1 1 = for j = 1, 2,...,ν setting = 0 κj kj − kj+1 kν+1 194 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY or equivalently, by~~

~~1 1 = kν κν  1 1 1  = +  k κ κ  ν−1 ν ν−1  .  . 1 1 1 1  = + + +  k1 κν κν−1 ··· κ1   Theorem 8.4.3 can be generalized as follows.~~

~~Theorem 8.4.5 (Balser-Tougeron: general case)~~

k-Li-summability on I and k-summability on I are equivalent with “same” sum. Precisely, the k-Li-sum of a series f(x) with k-sum (f1,f2,...,fν) on I is equal to fν and consequently, fν reads e

′ ′ ′ fν(x)= κ ,θ κ ,θ κ ,θ κ1 κν−1 κν f(x) L ν ν ◦···◦L 2 2 ◦L 1 1 ◦B ◦···◦B ◦B ◦ ′ ′ when the formula makes sense and, especially, for directions θ1,...,θe ν close to the bisecting direction θν of Iν and corresponding x.

Proof. — The theorem can be proved by recurrence as follows. It is trivially true for ν = 1 (and proved for ν = 2 in Theorem 8.4.3). Suppose it is true for ν 1 and prove it for ν. The fact that f(x) be k-Li-summable on I is now − equivalent to the fact g(ξ) be k-summable on I with k-sum (g1,g2,...,gν−1); and that moreover, gν−1 be defined withe exponential growth of order kν at infinity on Σ. The proofb thatb this is equivalentb to saying that f(x) is k- summable on I with sum (f ,f ,...,f ) satisfying f = (g ) is similar 1 2 ν ν Lkν ν−1 to the proof of Theorem 8.4.3 but the fact that the 1-cochain f1 ehas to be replaced by (f ,f ,...,f ) with jumps of order k on I I , k on 1 2 ν−1 ν ν−1 \ ν ν−1 I I ,...,k on I I . We leave the details to the reader. ν−2 \ ν−1 2 1 \ 2 Suppose by recurrence that the sum gν−1(ξ) of g(ξ) satisfies the formula of the theorem computed with values (k1,k2,...,kν) replaced by (k1, k2,... kν−1). Then, the associated values (κ1,κ2,...,κb ν−1) remain unchanged. Moreover, f is given by ′ (g ) (observe that k = κ ) and ν Lκν ,θν ν−1 ν ν theb formulab b follows. This ends the proof. 8.5. FOURTH APPROACH: BALSER’S DECOMPOSITION INTO SUMS 195

~~8.5. Fourth approach: Balser’s decomposition into sums~~

Suppose again that we are given a multi-level k = (k1,k2,...,kν) (cf. Def. 8.2.9) and a k-multi-arc I =(I1,I2,...,Iν). We saw in Proposition ν 8.2.14 (ii) that a sum j=1 fj(x) of kj-summable series on Ij is a k-summable series on I. We address now the converse question: P Do such splittingse characterize k-summable series on I?

~~The answer is yes when k1 > 1/2. Otherwise, one might have to introduce ramiﬁed series. The condition k1 > 1/2 is weakened in Theorem 8.6.7.~~

8.5.1. Case when k1 > 1/2. — Look ﬁrst at the relations between the various splittings of a given series. Proposition 8.5.1.— Splittings are essentially unique. Precisely, suppose the series f(x) admits two splittings f(x)= f (x)+ f (x)+ + f (x)= f ′ (x)+ f ′ (x)+ + f ′ (x) 1 2 ··· e ν 1 2 ··· ν where, for j = 1, 2,...,ν, the series f (x) and f ′(x) are k -summable on I . e e e e j e j e j e j Then, there exist series uj(x) such that, for j = 1, 2,...,ν, ′ e e fj(x)= uj(x)+ fj(x) uj+1(x) e − where u1 = uν+1 = 0 and, for j = 2,...,ν, the series uj is kj-summable e e e e on Ij−1. ′ ′ Moreover,e e the kj-sums fj of the fj’s and fj of the fj’se satisfy f (x)+ f (x)+ + f (x)= f ′ (x)+ f ′ (x)+ + f ′ (x) on I 1 2 ··· ν 1e 2 ··· e ν ν Notice that since uj(x) is kj-summable not only on the kj-wide arc Ij but on the kj−1-wide arc Ij−1 it is also kj−1-summable on Ij−1. e Proof. — The series u (x) = f ′ (x) f (x) is k -summable on I and, in ν ν − ν ν ν particular, is an sν-Gevrey series. Being equal to e e f (x)+e + f (x) f ′ (x)+ + f ′ (x) 1 ··· ν−1 − 1 ··· ν−1 it is also (k1,...,k ν−1)-summable on (I1,...,I ν−1). From the Tauberian The- e e e e orem 8.7.5 we deduce that uν(x) is kν-summable on Iν−1 and, a fortiori, is k -summable on I . Applying the same argument to the series f(x) f (x) ν−1 ν−1 − ν and to its two splittings e e e f (x)+ + f (x) and f ′ (x)+ + f ′ (x)+ f ′ (x)+ u (x)) 1 ··· ν−1 1 ··· ν−2 ν−1 ν proves the existence of uν−1(x) and we conclude to the existence of all uj’s by e e e e e decreasing recurrence. The equality of the sums follows directly. e e e 196 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

~~We can now give a new deﬁnition of multisummability as follows.~~

Deﬁnition 8.5.2.— Assume k1 > 1/2. ⊲ A series f(x) is said to be k-split-summable on I if, for j = 1, 2 ...,ν, there exist kj-summable series fj(x) on Ij such that e f(x)= f (x)+ f (x)+ + f (x). 1 e 2 ··· ν ⊲ The k-split-sume of f(xe) on I eis the functione f(x) uniquely deﬁned on Iν from any splitting of f(x) by e f(x)= f (x)+ f (x)+ + f (x), e 1 2 ··· ν where, for j = 1, 2,...,ν, fj(x) denotes the kj-sum of fj(x) on Ij.

Theorem 8.5.3 (Balser [Bal92a]).— Assume k1 >e1/2. A series f(x) is k-split-summable on I if and only if it is k-summable on I. Moreover, the k-split-sum and the k-sum agree. e Proof. — The “only if” part was considered in Remark 8.2.15 and above. Prove the converse assertion: if f(x) is k-summable on I then it is k-split- summable on I. Treat first the case when ν =e 2. Set s1 = 1/k1 and s2 = 1/k2 as usually. The series f(x) being s -Gevrey has a k -quasi-sum f (x) H0 S1; / ≤−k1 1 1 0 ∈ A A (cf. Def. 6.2.3) and, by hypothesis (cf. Def. 8.3.1), there exists f (x) 1 in H0 I ; e/ ≤−k2 such that f mod ≤−k1 = f on I and there ex- 1 A A 1 A 0 1 ists f (x) in H0 I ; such that f mod ≤−k2 = f on I . In other 2 2 A 2 A 1 2 words, f can be represented by a 0-cochain ϕ with values in / ≤−k1 0 0 A A and satisfying the following properties: its restriction ϕ1 = ϕ to I1 repre- 0|I1 sents f and has values in / ≤−k2 ; its restriction to I is the asymptotic 1 A A 2 function ϕ = f2. From Lemma 8.2.3 applied to f1(x) on I1 we are given 0|I2 f ′ (x) H0 I ; and f ′′(x) H0 S1; / ≤−k2 such that 1 ∈ 1 A 1 ∈ A A f = f ′ mod ≤−k2 + f ′′ on I . 1 1 A 1 1 ′′ ≤−k2 ′′ There exists then a 0-cochain ϕ1 with values in / representing f1 ′′ ′ A A ′′ which satisfies ϕ1 = ϕ1 f1 in restriction to I1. From Corollary 6.2.2, f1 (x) − ′′ ′′ can be identified to an s2-Gevrey series f1 (x) of which f1 (x) is a k2-quasi-sum. ′′ 0 ′′ In restriction to I2 the 0-cochain ϕ1 belongs to H (I2; ) since ϕ1| = f2 A I2 − ′ e ′′ f . Therefore, according to Definition 6.2.4, the series f1 (x) is k2-summable 1|I2 ′ ′ ′′ on I2 with k2-sum f2(x) f1| (x). Consider now the 0-cochain ϕ1 = ϕ0 ϕ1 − I2 e − 8.5. FOURTH APPROACH: BALSER’S DECOMPOSITION INTO SUMS 197 which belongs to H0 S1; / ≤−k1 and denote by f (x) the s -Gevrey series A A 1 1 it defines (cf. Cor. 6.2.2). The 0-cochain ϕ′ has no jump on I since 1 1 ′′ ′ e ′ ϕ0|I ϕ1| = ϕ1 (ϕ1 f1)= f1. 1 − I1 − −

And this, again by Definition 6.2.4, means that f1(x) is k1-summable on I1. We have thus proved that f(x) = f1(x)+ f2(x) where f1 is k1-summable on ′′ e I1 and f2 = f1 is k2-summable on I2. To prove the general casee one proceedse e by recurrence.e It suffices to prove that whene f(ex) is k-summable on I there exist a kν-summable series fν(x) on ′ ′ ′ ′ Iν and a k -summable series g(x) on I (where k =(k1,k2,...,kν−1) and I = (I1,I2,...,Ieν−1)) such that f(x)= g(x)+ fν(x). Indeed, let (f1,f2,...,fe ν) denote the k-sum of f(x) one I. The k1-quasi-sum f0(x) is now represented by a 0-cochain ϕ with valuese in e/ ≤−ke1 with the following properties: 0 A A for j = 1, 2,...,ν its restrictione ϕj to Ij represents fj on Ij; for j = 1, 2,...,ν −k − 1 the restriction to Ij has values in / j+1 and for j = ν the restriction to Iν A A ′ 0 is ϕ = fν. Apply Lemma 8.2.3 to fν−1 on Iν−1 to get f (x) H (Iν−1; ) 0|Iν ∈ A and f ′′(x) H0 S1; / ≤−kν such that ∈ A A f = f ′mod ≤−kν + f ′′ on I . ν−1 A ν−1 ′′ ′′ Like for f1 above, the section f determines a series fν(x) which is kν- summable on I . There exists a 0-cochain ϕ′′ with values in / ≤−kν which ν A A represents f ′′ and satisfies the condition ϕ′′ = ϕ f ′eon I . The 0- ν−1 − ν−1 cochain ϕ ϕ′′ shows that the series g(x)= f(x) f (x) is k′-summable 0 − − ν on I′. Hence, the result. e e e Remark 8.5.4. — One must be aware of the fact that the splitting strongly depends on the choice of the multi-arc of summation (on the direction of summation if all arcs are bisected by the same direction). It would be interesting to know which series admit a global splitting, i.e., the same splitting in almost all direction.

8.5.2. Case when k1 ≤ 1/2. — Choose r N such that rk > 1/2. ∈ 1 We know from Proposition 8.2.16 (i) that the series f(x) is k-summable r on I if and only if the series g(x) = f(x ) is rk-summable on I/r. We e ν can then apply Balser’s Theorem 8.5.3 to g(x) to write g(x)= j=1 gj(x) where the series gj(x) are rkj-summablee one I . This way, we obtain a split- j/r P ν 1/r ting f(x)= j=1 fj(x) of f(x) by setting fej(x)= gj(x e) for all j. How-e e ever, the seriesP fj(x) thus obtained are, in general, ramified series (in the e e e e e e 198 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY variable x1/r) and the splitting does not fit the statement of Theorem 8.5.3. Actually, as shown by the example below in the case when k1 = 1/2 there might exist no splitting in integer powers of x and we are driven to set the following definition.

Definition 8.5.5.— Suppose k 1/2. 1 ≤ A series f(x) is said to be k-split-summable on I if, given r N such r ∈ that rk1 > 1/2, the series g(x)= f(x ) satisfies Definition 8.5.2. e With this definition and Proposition 8.2.16 (i) we can assert in all cases e e the equivalence of k-summability and k-split-summability on I with “same” sum. Uniqueness holds as follows: ′ ′ r ′ Let r and r be such that rk1 and r k1 > 1/2. Set g(x) = f(x ) and g (x) = ′ f(xr ) with splittings g(x)= g (x)+ + g (x) and g′(x)= g′ (x)+ + g′ (x) 1 ··· ν 1 ··· ν respectively. Denote by R = ρr = ρ′r′ the l.c.m.e of re and r′.e Then, ρ ρ ′ ρ′ ′ ρ′ R ge1(x )+ + gν(x )e and g e(x )+ + eg (x ) aree two splittingse of f(ex ) ··· 1 ··· ν into Rk1-, ...,Rkν-summable series. Henceforth, they are essentially equal (ecf. Prop. 8.5.1).e e e e Show now that there might exist no splitting into integerl power series.

To this end, consider the case of a multi-level k =(k1,k2) satisfying 1/κ := 1/k 1/k 2. 1 1 − 2 ≥ Let (I1,I2) be a k-multi-arc. Assume, for instance, that I1 and I2 are closed with same middle point θ0 and, by means of a rotation, that θ0 = 0. For simplicity, assume that 1/κ < 4 and that I = π/k . Thus, with notations 1 | 2| 2 of Sections 6.3.3 and 8.4, the closed arc I centered at 0 with length I = 1 | 1| I π/k overlaps just once (since 2π I < 4π) and I reduces to θ = 0. | 1|− 2 ≤ | 1| 2 Consider a series f(x)= f1(x)+ f2(x)b which is the sum of a k1-summableb series f1(x) on I1 and of a k2-summable seriesb f2(x) onb I2 (series in integer powers of x). Denotee by eg = e(f), g = (f ) and g = (f ) the Bk2 1 Bk2 1 2 Bk2 2 series deducede from f, f1 and f2 by a k2-Borel transform. We know from Theorem 6.3.11 that g1(ξ) isbκ1-summablee b on I1 withe κ1-sumb g1(ξ).e The theorem asserts also thate e g1(ξ) hase an analytic continuation to an unlimited open sector Σ = J1 ]0b, + [ containing I1 ]0, +b [ with exponential growth × ∞ × ∞ • of order k2 at inﬁnity. Since Σ is wider than 2π it has a self-intersection Σ that contains theb negative real axis. Narrowingb it if necessary we can assume that Σ overlaps just once like I does. Denote by g• (ξ)= g (ξ) g (ξe2πi) 1 1 1 − 1 b 8.6. FIFTH APPROACH: ECALLE’S´ ACCELERATION 199

• the diﬀerence of the two determinations of g1 on Σ. On the other hand, g2(ξ) is convergent with sum g2(ξ). Hence, the diﬀerence of two determina- • 2πi tions g2(ξ)= g2(ξ) g2(ξe ) is identically 0 near 0 and can then be continued • −• • • allb over Σ by 0. Set g(ξ)= g1(ξ)+ g2(ξ). We can thus state: • Lemma 8.5.6.— The germ g•(ξ) can be analytically continued all over Σ. Proposition 8.5.7.— With notations and conditions as before (and especially, the condition 1/κ := 1/k 1/k 2) there exists series that are 1 1 − 2 ≥ (k1,k2)-summable on (I1,I2) but cannot be split into the sum of a k1-summable series on I1 and a k2-summable series on I2 if one restricts the splitting to series in integer powers of x.

Proof. — To exhibit a counter-example to the splitting of (k1,k2)-summable • n n series suppose that g( 1) = 0 and consider the series G(ξ)= g(ξ). n≥0( 1) ξ . − 6 • • − Set then G(ξ)= g(ξ)/(1 + ξ) and G(ξ)= g(ξ)/(1 + ξ) and denoteP the k2- Laplace transform of the series G(ξ) by F (x)= (Gb)(x). Theb function G(ξ) Lk2 is κ1-asymptotic to G(ξ) on I1 (cf. Prop.2.3.12) and it can be analytically continued with exponential growthb of ordere k2 at infinityb to an unlimited open sector σ containing I b ]0, + b[. Indeed, the function 1/(1 + ξ) is bounded at 2× ∞ infinity and has a pole at -1. The function g(ξ) is analytic with exponential ′ ′ growth of order k2 bat infinity on an unlimited open sector σ . In case σ does not contain the negative real axis one can take σ = σ′; otherwise, set σ = σ′ (ξ) > 0 for instance. According to Definition 8.4.1 and Balser- ∩ ℜ Tougeron Theorem 8.4.3 this shows that the series F (x) is (k1,k2)-summable • • on (I1,I2). However, G(ξ) = g(ξ)/(1 + ξ) has a pole at ξ = 1 which • • −• belongs to Σ and, thus, G(ξ) cannot be continued upe to infinity over Σ. From the lemma we conclude that the series F (x), however (k1,k2)-summable on (I1,I2), is not the sum of a k1-summable series on I1 and of a k2-summable series on I2 if one requires series in integere powers of x.

8.6. Fifth approach: Ecalle’s´ acceleration Historically, this approach called accelero-summation was the ﬁrst able to solve the problem of summation in a case of several levels. First introduced by J. Ecalle´ in a very general setting applying to series solutions of non-linear equations and more general functional equations, it was adapted by J. Mar- tinet and J.-P. Ramis to the case of solutions of linear diﬀerential equations in [MR91]. The method proceeds by recursion on increasing levels whereas 200 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY the iterated Laplace approach runs with decreasing levels. Each step is performed with the use of special integral operators called accelerators or Ecalle’s´ accelerators which involve the successive levels taken two-by-two. In this section, for simplicity, we work in a given direction θ0, i.e., we consider only multi-arcs I =(I1,I2,...,Iν) with common middle point θ0. Begin by observing what happens on the example of the Ramis-Sibuya series RS(x) (Exa. 8.1.1).

Examplef 8.6.1 (accelero-summation of RS(x)).— We saw in Exam- ple 8.1.1 that the series RS(x) is k-summable for no k > 0 in the directions θ ∈ [π/4 , 3π/4] mod π and therefore, no k-Borel-Laplaceg process applies in these directions. In the case of RS(x) and moref generally of a (1, 2)-summable series the method consists, in some way, in applying simultaneously a 1- and a 2-Borel-Laplace process as shown below. Fix a nonf anti-Stokes direction θ belonging to ]π/4 , 3π/4[ mod π and, when no

confusion is possible, denote simply by 1 and 2 instead of 1,θ and 2,θ the 1- and B B B B the 2-Borel transforms in direction θ and by 1 and 2 instead of 1,θ and 2,θ the L L L L 1- and the 2-Laplace integrals in direction θ (cf. Def. 6.3.5). Contrary to the 2-Borel transform the (formal) 1-Borel transform applied to E(x) and L(x) provides convergent series. This invites us to begin with the 1-Borel-Laplace process followed by the 2-Borel- Laplace process. The 1-Borel transform of E(x) can bee continuede to inﬁnity in direction θ (and neighboring directions) with exponential growth of order one and can then be applied

~~a Laplace operator 1. On the contrary, the 1-Borel transform of L(x) can only be L continued with exponential growth of order two (cf. Exa. 6.3.14). Hence, the Laplace~~

~~operator 1 does not apply to 1 RS(x) (ξ). L B A solution to this problem consists in merging the next two arrows of the process as indicated in the diagram: f~~

~~Formally, we can write~~

A2,1(ϕ)(ζ) = 2 1(ϕ) (ζ) B L eiθ ∞ 1 2 1/2 dt = eζ /t ϕ(ξ)e−ξ/t dξ 2πi t2 γ2θ 0 Z iθ Z 1 e ∞ ζ2 ξ dt = ϕ(ξ) exp dξ (commuting the integrals) 2πi t 1/2 t2 0 γ2θ − t Z eiθ ∞ Z 1 ξ 1/2 du 2 = ϕ(ξ) exp u u 2 dξ (setting u = ζ /t) 2πi 0 H − ζ ζ Z Z where denotes a Hankel contour around the negative real axis. Setting u1/2 = iv in H 1 u−τu1/2 the integral kernel 2(τ) = 2πi H e du we recognize the derivative of the Fourier C −v2 −τ 2/4 transform of the Gauss function e and we obtain 2(τ)= τe /(2√π). This kernel R C 8.6. FIFTH APPROACH: ECALLE’S´ ACCELERATION 201

is (for τ > 0) exponentially small of order two and can then be applied to the 1-Borel ℜ transform ϕ(ξ) of the Ramis-Sibuya series RS(x). The operator A2,1 deﬁned by iθ 1 e ∞ A2,1 ϕ(ξ) (ζ)= f ϕ(ξ) 2 ξ/ζ dξ ζ2 C Z0 is called (2, 1)-accelerator.

Now, the function A2,1 ϕ(ξ) (ζ) satisﬁes the same equation as 2(RS(x))(ζ) and B can thus be applied a 2-Laplace transform (see Exa. refRSsum3). Finally, we obtain an asymptotic function on a quadrant bisected by θ. This function has thef Ramis-Sibuya series RS(x) as Taylor series since, formally, the followed process is the identity.

f The formal calculation made above to define the accelerator A2,1 can be made with A = (ϕ) for any pair of levels k 1. Cα 2πi ZH When it is useful to make explicit the direction θ in which the integral is taken θ we denote Ak2,k1 . Proposition 8.6.2.— Given α> 1 let β denote its conjugate number: 1/α + 1/β = 1. The kernel (τ) is flat of exponential order β at infinity on the sec- Cα tor arg(τ) < π/(2β), i.e., for all δ > 0 there exist constants c ,c > 0 such | | 1 2 that (τ) c exp( c τ β) on arg(τ) π/(2β) δ. |Cα | ≤ 1 − 2 | | | | ≤ − Proof. — The proof was already given in the part “(i) implies (ii) point 3” of the proof of the pre-Tauberian Theorem 6.3.11. To use the same notations, α perform the change of variable U = τ w in the integral defining Cα(τ). Set 1/α α 1/α F (U) = exp( U ), ζ = 1/τ and G(ζ)= Cα(ζ )/ζ. We obtain G(ζ) = − ζU 1/(2πi) H F (U)e dU. Then, set k1 = 1, k2 = α and κ1 = β. Set an = 0 for all n in (42) (the exponential F (U) is flat on arg(U) < απ/2). We conclude R from Estimate (41) and Proposition 2.3.17. Corollary 8.6.3.— Denote, as before, 1/κ = 1/k 1/k . 1 1 − 2 The accelerator Ak2,k1 applies to any function ϕ with exponential growth of order κ1 at infinity in direction θ. 202 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

With this result the accelerator Ak2,k1 appears like similar to a κ1-Laplace operator and has similar properties. Let us state the following result generalizing Theorem 6.3.11 “(ii) = (i)” (cf. [Bal94, Thm 5.2.1]). ⇒ Lemma 8.6.4.— Let k, k1, k2 > 0 be given. Assume k1 < k2 and define κ and k by 1 1 1 1 1 1 =b and = + . κ k1 − k2 k κ k Let σ = I ]0, + [ be an unlimited sector and denote by I the arc with × ∞ same middle point θ as I and length I + π/κ. b 0 | | Supposeb thatb g(ξ) is analytic on σ, it belongs to (I) at 0 and it has A1/k exponential growth of orderb κ at infinityb on σ. b Then, the function f(ξ)= A (bg)(ξ) belongs to (bI). k2,k1 A1/k b Definition 8.6.5.— Let k =(k1,k2,...,kν) be a multi-level and θ0 a direction. A series f(x) is said to be k-accelero-summable (or, accelero-summable) in direction θ0 if it can be applied the following sequence of operators in direction θ = θ0 ande neighboring ones resulting in the accelero-sum f(x): A A A Bk1 k2,k1 k3,k2 kν−1,kν Lkν f(x) • • • • • f(x). −−−→ −−−→ −−−→ ··· −−−→ −−−→ By the expression “can be applied” we mean that the kernels of the integral e operators have, at each step, the right growth rate at infinity for the integral to exist. The term accelero-summation is commonly used for a larger class of operators associated with various kernels depending on the type of problem one wants to solve. Here, we refer always to the definition given above. Theorem 8.6.6.— k-multisummability and k-accelero-summability in a given direction θ0 are equivalent with “same” sum. Precisely, if (f1,f2,...,fν) is the k-multisum of a series f(x) in direction θ0 then, f = fν is its k-accelero-sum in direction θ0. e Proof. — We sketch the case of two levels k = (k1,k2) letting the reader perform the general case by iteration. Since the theorem holds true for polynomials we can assume that the given series have valuation greater than k2 so that their k2-Borel series contain only positive powers of ξ. ⊲ multisummability implies accelero-summability. Without loss of generality we can assume that k1 > 1/2. From Theorem 8.5.3 it is then sufficient to prove that k1- and k2-summable series in direction θ0 are accelero-summable in the same direction. Suppose f(x) is k1-summable

e 8.6. FIFTH APPROACH: ECALLE’S´ ACCELERATION 203 in direction θ with k -sum f (x). Then, (f)(ξ) is convergent at 0 and its 0 1 1 Bk1 sum ϕ(ξ) can be analytically continued with exponential growth of order k1 at infinity in direction θ0 and neighboring ones. Sincee κ1 >k1 one can apply the accelerator Ak2,k1 to ϕ(ξ) in direction θ0 and neighboring ones and the resulting function can be analytically continued to infinity with moderate growth; it can then be applied a k2-Laplace transform to produce a (k1,k2)-sum f(x). Actually in that case, A (ϕ) = (ϕ) (the integrals commute). It k2,k1 Bk2 ◦Lk1 follows that f(x)= (f )(x) and f(x) is the restriction of f to an Lk2 ◦Bk2 1 1 open sector with opening larger than π/k2 (but possibly smaller than π/k1) centered at θ0. Suppose f(x) is k2-summable in direction θ0 with k2-sum f(x) and let κ1 be defined by 1/κ = 1/k 1/k . The k -Borel transform g (ξ)= (f)(ξ) 1 1 − 2 1 1 Bk1 of f(x) definese an entire function g1(ξ) with exponential growth of order κ1 at e infinity. Lemma 8.6.4 applied to g1(ξ) with 1/k = 0 shows thatb Ak2,k1 (g1)(ξ) is κ -asymptotice to g (ξ)= (f)(ξ) on a sector of opening larger than π/κ bi- 1 2 Bk2 1 sected by θ0. Since f(x) is k2-summable in directionb θ0 its k2-Borel series g2(ξ) e is convergent andbAk2,k1 (g1)(ξ) coincide with the sum g2(ξ) of series g2(ξ). A k2-Laplace transforme provides then the k2-sum f(x) of f(x) in directionbθ0. ⊲ Accelero-summability implies multisummability. b e Suppose f(x) is (k1,k2)-accelero-summable in direction θ0. Thus, its k1-Borel p transform g(ξ)= cpx converges on a disc Dρ = ξ < ρ . Its sum g(ξ) p>k2 {| | } can be analyticallye continued to an open sector σ neighboring the direction θ P 0 with exponentialb growth of order κ1 at infinity (recallb 1/κ1 = 1/k1 1/k2). − We choose σ so narrow about θ0 that f(x) isb (k1,k2)-accelero-summable in all direction θ belonging to σ. Without loss of generality, we assume that 1/κ1 < 2.b Show now that under thate condition the series f splits into a sum f(x)= f1(x)+ f2(x) whereb f1(x) and f2(x) are k1- and k2-summable in direction θ0 respectively (cf. [Bal92a, Lem. 1 and 2]). e Showe firste thate it suffices toe considere the case when f(x) is (k1,k2)- summable (i.e.,(k1,k2)-summable in almost all direction). Let 0 < r < ρ. The circle γ centered at 0 with radius r belongs to Dρ. Denotee by γ1 the arc of γ oriented positively outside the sector σ and by γ2 the arc oriented positively inside σ. From Cauchy’s integral formulab we know that, on the interior Dr of γ, the function g(ξ) satisfies b b b 1 g(η) g(ξ)= dη 2πi η ξ Zγ − 204 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

and g(ξ)= g1(ξ)+ g2(ξ) there if one sets 1 g(η) 1 g(η) g (ξ)= dη and g (ξ)= dη. 1 2πi η ξ 2 2πi η ξ Zγ1 − Zγ2 − The function g (ξ) has an analytic continuation to D σ which is bounded 1 r ∪ at inﬁnity. The k1-Laplace transform f1(x) of the Taylor series g1(ξ) of g1(ξ) is therefore a k1-summable series in direction θ0. Denoteb byb cℓ(σ) the closure of σ and set σ′ = C cℓ(σ). Thus, C = bσ σ′ d′ d′′ where d′ andb d′′ are the \ ′ ∪ ∪ ∪ two half-lines limiting σ and σ . Similarly to g1(ξ), the functionbg2(ξ) has an ′ analyticb continuationb tobDr σ whichb is boundedb at inﬁnity. Its k1-Laplace ∪ ′ transform f2(x) is thusbk1- henceb (k1,k2)-summable in all directions of σ . On another hand, f (x)= f(xb) fb(x), is (k ,k )-summable in all directions of σ. 2 − 1 1 2 We concludeb that f2(x) being (k1,k2)-summable in all directions but, maybe,b ′ ′′ the singular directionsb db and bd , is (k1,k2)-summable. To continue the proofb we can then assumeb that f(x) is (k1,k2)-summable (in almost all direction).

~~Figure 5~~

Prove now the followed splitting f(x) = f1(x)+ f2(x) where f1(x) and f2(x) are respectively k1- and k2-summable in direction θ0. From Lem. 8.6.4 θ θ e e e e the function h (ζ)= Ak2,k1 (g)(ζ) is κ1-asymptotic to a same series (precisely, e to the k2-Borel transform of f) on a sector of size π/κ1 in all non-singular direction θ. These functions are analytic continuations from each others as long as θ does not pass a singulare direction. With f(x) we can thus associate a 0-cochain (ϕ (ζ)) with ϕ (U ) all asymptotic to the same j j∈J j ∈ A1/κ1 j series f(ζ)= (f)(ζ), the sectors U having openinge U > π/κ and mak- Bk2 j | j| 1 ing a good covering = (U ) of a punctured neighborhood of 0 in C U j j∈Z/pZ (cf. Def.b 3.2.9). Wee can choose the covering such that U is bisected by θ . U 0 0 Denoting by θj the direction bisecting Uj observe that, under such conditions, there might be several singular directions between θj and θj+1. Notice that such a covering is made possible due to the condition 1/κ1 < 2. Denote, as 8.6. FIFTH APPROACH: ECALLE’S´ ACCELERATION 205

• • previously, by U j= Uj Uj+1 the nerve of . For all j Z/pZ choose aj U j ∩ U ∈ ∈ and apply the Cauchy-Heine Theorem (Thm. 2.5.2) to build a new 0-cochain • with associated 1-cocycle (ϕj= ϕj ϕj+1). The construction is as follows. De- − ′ compose the 1-cocycle (ϕj) into the sum of the elementary 1-cocycles ϕj = ϕj • • • ′ • ′ ′ on U j and 0 on U ℓ when ℓ = j. Set r = min( aj ) and U j=U j ζ < r . 6 • ′| | ∩{| | } ′ Denote by Uj the sector with self intersection U j. The Cauchy-Heine Theorem (Thm. 2.5.2) says that the function

• 1 aj ϕ (t) ψ′ (ζ)= j dt j 2πi t ζ Z0 − ′ ′ ′ can be analytically continued to Uj with 1-cocycle ϕj(ζ) and ψj(ζ) is κ1-Gevrey • m aj ϕ m+1 asymptotic to the series cmζ where cm = 1/(2πi) 0 j (t)/t dt. It has also an analytic continuation to C deprived of the half-line d =]0,α [, α = P R j j j arg(aj) and it tends to 0 at infinity. Define the analytic function ψj(ζ) on Uj by setting ψ (ζ)= ψ′ (ζ), ζ U j ℓ ∈ j ℓ∈XZ/pZ ′ (choose the determinations of the ψℓ that are analytic on all of Uj). Suppose now that a and a have been chosen so that the angle α α is > 0 p−1 | 0 − p−1| π/κ1 and bisected by θ0. This is possible since the opening of U0 is larger than π/(2κ1) on both sides of θ0. Denote by V0 the unlimited open sector ]α ,α [ ]0, [ and by Ψ (ζ) the analytic continuation of ψ (ζ) to V . The p−1 0 × ∞ 0 0 0 sector V0 is κ1-wide; the function Ψ0(ζ) has a κ1-asymptotic expansion Ψ0(ζ) at 0 and an exponential growth of order less than k2 at infinity on V0. Denote by f1(x) the k2-Laplace transform of the series Ψ0(ζ). It follows from Theoremb 6.3.11 (ii)= (i) that the series f (x) is k -summable in direction θ . ⇒ 1 1 0 eOn another hand, denote by φ0(ζ) theb asymptotic series of ϕ0 (it is actually the k2-Borel transforme of f(x)). By hypothesis, one can apply a -Laplace transform tobϕ in direction θ and neighboring Lk2 0 0 ones. This means that ϕ0 has an analytice continuation to an unlim- ′ ited sector V0 containing the direction θ0 with exponential growth of order k2. The 0-cochains ϕj(ζ) and ψj(ζ) induce the same 1-cocycle • ′ on (U ) . It follows that ϕj(ζ) ϕj+1(ζ)= ψj(ζ) ψj+1(ζ) for all j j∈Z/pZ − − j and the functions ϕ (ζ) ψ (ζ) glue together into an analytic function on j − j the disc D′ = ζ

ϕ (ζ) ψ (ζ) can be continued into an analytic function on V ′ D′ with 0 − 0 0 ∪ Taylor series (f )(ζ) and it has exponential growth of order k at inﬁnity. Bk2 2 2 This means that the series f2(x) is k2-summable in direction θ0. Moreover, f(x)= f1(x)+ f2(ex) and the result follows. e e e e The second part of the proof of Theorem 8.6.6 provides the following improvement of Theorem 8.5.3:

~~Theorem 8.6.7 (Balser [Bal93]).— Let be given a multi-level k =(k1,k2) and a k-multi-arc I =(I ,I ). Denote 1/κ = 1/k 1/k . Under the condi- 1 2 1 1 − 2 tion~~

~~κ1 > 1/2 then, k-split-summability and k-summability on I are equivalent with same sum. The property extends to multi-arcs I =(I ,...,I ) of length ν 2 under 1 ν ≥ the conditions~~

~~κ > 1/2 where 1/κ = 1/k 1/k for j = 1,...,ν 1. j j j − j+1 − Observe that the counter-example in Proposition 8.5.7 corresponds to ν = 2 and κ1 = 1/2.~~

~~8.7. Sixth approach: wild analytic continuation~~

8.7.1. k-wild-summability. — Like Ramis-Sibuya definition of k-summability was translated in terms of analytic continuation in the infinitesimal neighborhood Xk of 0 endowed with the sheaf k (cf. Sect. 6.4.1) Malgrange-Ramis F definition of k-multisummability can be translated in terms of analytic continuation in the infinitesimal neighborhood Xk of 0 endowed with the sheaf k (cf. Sect. 4.5.3). F

Deﬁnition 8.7.1 (k-wild-summability).— Let k = (k1,...,kν) be a multi-level and let I =(I1,...,Iν) be a k-multi-arc (cf. Def. 8.2.9). Set k , 0 =+ (cf. Nots. in Sect. 4.5.2). { ν+1 } ∞ 8.7. SIXTH APPROACH: WILD ANALYTIC CONTINUATION 207

n ⊲ A series f(x) = n≥0 an x is said to be k-wild-summable on I if it can be wild analytically continued in the infinitesimal neighborhood (Xk, k) P F of 0 to a domaine containing the closed disc D(0, k , 0 ) and the sec- { 1 } tors I ]0, k , 0 ] for all j = 1,...ν . j× { j+1 } ⊲ Its sum is the germ of analytic function defined on Iν by this wild analytic continuation. It is said to be k-wild-summable in direction θ if all arcs Ij are bisected by the direction θ. ⊲ The series is said to be k-wild-summable if it is k-wild-summable in almost all direction, i.e., all direction but finitely many called singular directions.

~~k1,k2 Figure 6. Domain for a (k1,k2)-sum in X (in white)~~

It follows from the Relative Watson’s Lemma 8.2.1 and Watson’s Theorem 6.1.3 that the continuation, hence the sum in the sense of wild-summation, when it exists, is unique. Since Deﬁnition 8.7.1 exactly translates Malgrange-Ramis deﬁnition of multisummability (Def. 8.3.1) we can state:

~~Proposition 8.7.2.— k-wild-summability is equivalent to k-summability in any of the previous sense with same sum.~~

Deﬁnition 8.7.3.— Let I be a k-multi-arc. ⊲ A sector built on I like in Deﬁnition 8.7.1 (cf. Fig. 6) is called a k-sector in Xk. 208 CHAPTER 8. SIX EQUIVALENT APPROACHES TO MULTISUMMABILITY

⊲ A kj-arc Ij such that f(x) can be wild analytically continued to the open sector I ]0, k , 0 [ but not to the closed sector I ]0, k , ] in Xk is said j× { j } j× { j ∞} to be a singular arc of level kje(for f(x)); otherwise it is said to be non singular. ⊲ A direction θ bisecting one or several singular arcs is said to be a singular direction for f(x); otherwisee it is said to be non singular.

~~From the viewpointe of wild analytic continuation the following results are straightforward.~~

Proposition 8.7.4.— Let k =(k1,k2,...,kν) be a multi-level. ⊲ A series is k-summable if and only if it admits finitely many singular arcs in Xk. ′ ⊲ Let k be a multi-level containing all levels k1,k2,...,kν of k. A series which is k-summable in a direction is also k′-summable in that direction. In other words, k-summability is stronger than k′-summability. Proof.— ⊲ If there is finitely many singular arcs then the series is k-summable in all directions but the finitely many bisecting directions of the singular arcs. Conversely, suppose that the series has infinitely many singular arcs. If k contains several levels then there is at least one level supporting infinitely many singular arcs and all bisecting directions of these arcs are singular directions for the series. Hence, the non-summability of the series. ⊲ From the viewpoint of wild analytic continuation the domain one has to continue the series towards its k-summability in direction θ contains the domain one has to continue it towards its k′-summability in the same direction.

8.7.2. Application to Tauberian Theorems. — The Tauberian The- orems 6.3.12 and 6.3.13 are easily generalized to multisummable series (cf. [MR91, Prop. 8 p. 349]). Without loss of generality we assume that the smallest level k1 is greater than 1/2.

~~Theorem 8.7.5.— Let k = (k1 ...,kν) be a multi-level, I be a k-multi-arc and k = . Suppose k′ satisﬁes k k′~~

Theorem 8.7.7.— With notations as before let κ = k′ k′′ be the multi-level ′ ′′ ∩ defined on the common values of k and k : κ =(κ1,...κν0 ) satisfies ′ ′ ′ ′′ ′′ ′′ κ ,...,κ = k ,k ,...,k ′ k ,k ,...,k ′′ . { 1 ν0 } { 1 2 ν } ∩ { 1 2 ν } A series f(x) which is both k′- and k′′-summable satisfies the following properties: (i) if κ =e then f(x) is convergent; ∅ (ii) if κ = then f(x) is κ-summable. 6 ∅ e ′ ′′ Proof. — (i)Casewhene κ is empty. Suppose for instance that k and k satisfy ′ ′ ′′ ′′ ′ ′ ′′ k1 <

ℓ 1 series; the Ij ’s form a cyclic covering of S and the consecutive intersections Iℓ Iℓ+1 are made of arcs of length larger than π/k . From the Ramis-Sibuya 1 ∩ 1 1 Corollary 6.2.2 and the Relative Watson Lemma 8.2.1 we conclude like in Section 6.4.2 that the corresponding sums glue together into a section of k F over the closed disc D with radius k , 0 . The series is then 1/k -Gevrey k2 { 2 } 2 and, by Theorem 8.7.5, it is (k2,...,kν)-summable. ′ ′′ The series f(x) being both k -summable and 1/k1 -Gevrey we know ′′ ′ ′ from Theorem 8.7.5 that it is at worst (k ,k ′ ,...,k ′ )-summable. As a 1 j1 ν ′ e ′′ k -summable series it has then no singular arc of level

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~~INDEX OF NOTATIONS~~

⋐ proper inclusion of sectors 5 sheaf over S1 of germs of asymptotic functions 47 A subsheaf of of s-Gevrey germs 47 As A <0 subsheaf of of flat germs, i.e., asymptotic to 0 47 A A ≤−k subsheaf of <0 of germs exponentially flat of order k 47 A A formal Borel transform 125 B Borel transform in direction θ 125 Bθ k-Borel transform in direction θ 129 Bk,θ Riemann surface of logarithm 5 C[[x]] differential algebra of formal power series with complex coefficients and derivation d/dx C x differential sub-algebra of C[[x]] restricted to convergent series { } C[[x]]s differential sub-algebra of C[[x]] restricted to Gevrey series of order s, i.e., of level k =1/s 18 C x differential sub-algebra of C[[x]] of k-summable series on I 116 { }{k,I} C x differential sub-algebra of k-multisummable series on I 181 { }{k,I} E(x) Euler series 8 E(x) Euler function 8 e homogeneous Euler operator 49 E0 Ei(x) exponential integral function 10 , presheaf and associated sheaf 42 F F 3F0 example of a hypergeometric series 11 −4 g(z) g(z)= z 3F0( 3, 4, 5 1/z) 11 { }| h(z) example of a solution of a mild difference equation 13 e ℓe(z) example of a solution of a wild difference equation 14 ek,k1,... levels (positive numbers) e 218 INDEX OF NOTATIONS k,I multi-level and multi-arc 180 Laplace transform in direction θ 125 Lθ k-Laplace transform in direction θ 129 Lk,θ s,s1,... orders, i.e., inverses of levels s =1/k,s1 =1/k1,... α,β(R) open sector x ; α< arg(x) <β and 0 < x

Asymptotics Fine Borel-Laplace summation, k-fine Poincaréasymptotics, 6 summability, 132 Gevrey asymptotics, 17 Formal fundamental solution, 72, 81 Basic transformations Germs of diffeomorphism k-Borel and k-Laplace transforms, 28, 129 Birkhoff-Kimura theorem, 163, 166 Borel and Laplace transforms, 125 conjugacy, 159 Truncated k-Laplace transform, 29 invariance, 167 Borel-Ritt summability, 169 application, 31, 88, 114, 166 Homological system, 75, 90 theorem, 26, 30, 51, 58 Index, irregularity Cauchy-Heine definition, 99, 100 application, 104 example with no index, 101 theorem, 103, 105, 110 integral, 31 Infinitesimal neighborhoods theorem, 32, 61 application, 109, 155 Cyclic vector lemma, 64, 68 big point, 93, 96, 110, 157 Deligne-Malgrange theorem, 103 definition, 90, 93, 96 Determining polynomial, 75 Maillet-Ramis theorem, 108, 121 Differential module Main asymptotic existence theorem D-module, 65, 67, 69 application, 78, 102, 104, 121, 122 connection, 65 theorem, 88, 89 Definition, 65, 68 Malgrange-Sibuya theorem, 78 equivalence of equations, 70 Newton polygon equivalence of systems, 66, 68 characteristic equation, 86 Examples definition, 81, 82 Euler, 8, 18, 21, 30, 35, 48, 59, 60, 83, 89, indicial equation, 86 103, 113, 115, 116, 123, 141 N. polygon & Borel transform, 84 exponential integral, 10, 21 Presheaves hypergeometric, 11, 18, 22, 83, 89, 116, 123 A, 7, 15, 16 lacunar series, 46 As, 21 Leroy, 150 A<0, flat functions, 7, 15 mild, 13, 18, 22, 116, 142 A≤−k, 25 Ramis-Sibuya, 171, 179, 184, 187, 200 definition, 37 wild, 14, 18, 22, 116, 142 morphism, 38 220 INDEX

Ramis-Sibuya theorem, 119 sheaf As, 47 Resurgence, summable-resurgence, 153 support of a section, 53 Sectors, arcs Stokes phenomenon k-wide sector, k-wide arc, 110, 113, 115 anti-Stokes directions, 75, 78 Open or closed sector, 5 application, 78, 102 Proper sub-sector, 6 example, 35 Sheaves levels, 75 definition, 41 size of sectors of summation, 89 direct image, 51 Stokes arcs, 74, 105 espace étalé, 42 Stokes cocycle, 79 extension by 0, 53 Stokes cocycle vs sums, 80 morphism, 44 Stokes directions, 76, 102, 104 quotient sheaf, 48 Stokes matrices, 80, 81 restricted sheaf, 52 Stokes values, 152, 154 sheaf A, 47 Watson sheaf A<0, flat germs, 47 relative Watson’s lemma, 175 sheaf A≤−k, 47 Watson’s lemma, 114